Abstract
In this paper we investigate the global dynamics in an HIV virus infection model with saturated incidence. The model includes two viral strains, one is wildtype (i.e. drug sensitive) and another is drugresistant. The wildtype strain can mutate and become drugresistant during the process of reverse transcription. The nonnegativity and boundedness of solutions are established. The basic reproduction numbers of two strains and the existence of equilibria are also obtained. The threshold criteria on the local and global stability of equilibria and the uniform persistence of the model are established by using the linearization method, constructing suitable Lyapunov functions and the theory of persistence in dynamical systems. Moreover, the mathematical analysis and numerical examples show that model may have a positive equilibrium which is globally asymptotically stable.
Introduction
It is well known that mathematical models that describe the dynamical behaviors of virus infection play an important role in understanding the mechanism of the diffusion of virus. There has been much interest in mathematical modeling of viral dynamics withinhost. So, the research of virus dynamics with specific immune response, which can control the virus propagation, has drawn significant attention [1–6]. A few years ago, Perelson et al. in [7] constructed a model that has been widely adopted to model the plasma viral load in HIV infected patients as follows:
Treating HIVinfected patients with a combination of several antiretroviral drugs usually contributes to a substantial decline in viral load and an increase in \(\mathit{CD}_{4}^{+}\) T cells. Nevertheless, there is a reasonable chance that drugresistant variants of HIV preexist even before the initiation of therapy due to a single mutation, or a number of mutation combinations can result in drug resistance by Ribeiro and Bonhoeffer in 2000 (see [8, 9]). In order to study the mechanism of the emergence of drug resistance during the treatment of HIVinfected patients, a dynamical model including wildtype and drugresistant strains was proposed by Rong et al. in [9] as follows:
Usually the rate of infection in most HIV1 models is assumed to be bilinear in the virus and the uninfected cells. However, the actual incidence rate is probably not linear over the entire range of virus and the uninfected cells. Thus, it is reasonable to assume that the infection rate of HIV1 is given by the Beddington–DeAngelis functional response [10], which was introduced by Beddington [11] and DeAngelis et al. [12]. For a specific nonlinear incidence rate, we consider the following HIV1 infection model with saturated incidence:
The biological significance of variables and parameters in model (2) is given in Table 1.
In model (2), the parameter u (\(0< u<1\)) is the conversion fraction at which cells infected by the wildtype mutate and become drugresistant during the process of reverse transcription of viral RNA into proviral DNA (SR conversion, for short). It should be noted that the backward mutation from drugresistant to wildtype strain is neglected since the wildtype virus dominates the population before the initiation of therapy (see [13, 14]). And the terms \(\frac{k_{s}V_{s}T}{1+\omega _{1}V _{s}}\) and \(\frac{k_{r}V_{r}T}{1+\alpha _{1}V_{r}}\) express the saturated incidence for virus \(V_{s}\) and \(V_{r}\), where \(\omega _{1}\) and \(\alpha _{1}\) are the nonnegative constants. When \(\omega _{1}=0\) or \(\alpha _{1}=0\), the corresponding incidence degrades into bilinear incidence for \(V_{s}\) or \(V_{r}\).
In [9], we see that model (1) with bilinear incidence is investigated. The authors only obtained the existence and local stability of the infectionfree equilibrium, the equilibrium with only wildtype virus, drugresistant virus, and the coexistence equilibrium (see Proposition 1 and Proposition 2 in [9]). We all know that in many realistic infectious diseases the nonlinear incidence rates play very important roles, and the global dynamics of the model, including the global asymptotic stability of equilibria, the uniform persistence, etc., also needs to be investigated in detail. In [5, 15], we see that the global dynamics for virus infection models with nonlinear incidence rates is discussed. Therefore, in this paper we carry out the research for a wildtype and drugresistant HIV infection model with saturated incidence. We establish a series of threshold criteria for the local and global asymptotic stability of infectionfree, drugresistant strain infection equilibria, and the uniform persistence of HIV infection.
The organization of this paper is as follows. In Sect. 2, the nonnegativity and boundedness of solutions are established, and then the basic reproduction numbers of two strains and the existence of equilibria are obtained. In Sect. 3, the main theorems on the local and global stability of equilibria of model (2) are stated and proved. In Sect. 4, the uniform persistence of model (2) is also investigated. In Sect. 5, some numerical examples are given to illustrate our main results. In the last section, a brief conclusion is presented.
Preliminaries
For any integer \(n>0\), denote \(R^{n}_{+}=\{(x_{1},x_{2},\ldots,x_{n}) \in R^{n}: x_{i}\geq 0, i=1,2,\ldots,n\}\). The initial condition for model (2) is given by
Firstly, on the positivity and boundedness of solutions for model (2), we have the following result.
Theorem 1
The solution \((T(t),T_{s}(t),V_{s}(t),T_{r}(t),V_{r}(t))\)of model (2) with initial condition (3) is defined for all \(t\in [0,\infty )\)and is nonnegative and ultimately bounded.
Proof
On the nonnegativity of solutions, by the continuity of solutions with respect to initial values, we only need to prove that, for any positive initial value \((T_{0},T_{s0},V_{s0},T_{r0},V_{r0})\), the solution \((T(t),T_{s}(t),V_{s}(t),T_{r}(t),V_{r}(t))\) with initial condition (3) is also positive for any \(t>0\) in the definition interval. From the first equation of model (2), we have
Hence, as \(T_{0}>0\), we directly have \(T(t)>0\) for any \(t>0\) in the definition interval.
Define \(m(t)=\min \{T_{s}(t),V_{s}(t),T_{r}(t),V_{r}(t)\}\). Obviously, \(m(0)=\min \{T(0),T_{s}(0),V_{s}(0), T_{r}(0),V_{r}(0)\}>0\). By the continuity of solutions there exists \(\delta >0\) such that \(m(t)>0\), when \(t\in [0,\delta )\). We only need to prove \(m(t)>0\) for all \(t\geq 0\) in the definition interval. Suppose that there exists \(t^{*}>0\) such that \(m(t^{*})=0\) and \(m(t)>0\) for all \(t\in [0,t^{*})\). Then there exist the following cases: (1) \(m(t ^{*})=T_{s}(t^{*})\), (2) \(m(t^{*})=V_{s}(t^{*})\), (3) \(m(t^{*})=T _{r}(t^{*})\), and (4) \(m(t^{*})=V_{r}(t^{*})\).
For case (1), according to \(m(t)>0\) for all \(t\in [0,t^{*})\), from the second equation of model (2), we know \(\frac{\mathrm {d}T_{s}(t)}{\mathrm {d}t}> \delta T_{s}\). Thus, \(T_{s}(t)>T_{s}(0)e^{\delta t}\) for any \(t\in [0,t^{*})\). Taking \(t \rightarrow t^{*}\), then \(0=T_{s}(t^{*}) \geq T_{s}(0)e^{\delta {t^{*}}}>0\), which leads to a contradiction. Similarly, we can get the contradiction for cases (2), (3) and (4). Therefore, \((T(t),T_{s}(t),V_{s}(t),T_{r}(t),V_{r}(t))\) is positive for all \(t\geq 0\) in the definition interval.
Define a Lyapunov function
We have
where \(n=\min \{d,\frac{\delta }{2},c\}\). Since solution \(U(t)\) of the comparison equation
with initial condition \(U(0)=U_{0}\geq 0\) is defined for all \(t\in [0,\infty )\) and satisfies \(\lim_{t\rightarrow \infty }U(t)=\frac{ \lambda }{n}\), by the comparison principle, we directly have that \(W(t)\) is bounded, and hence solution \((T(t),T_{s}(t),V_{s}(t),T_{r}(t),V _{r}(t))\) is also bounded. Thus, \((T(t),T_{s}(t),V_{s}(t),T_{r}(t),V _{r}(t))\) can be defined for all \(t\in [0,\infty )\). Furthermore, since \(W(t)\leq U(t)\) as \(W(0)\leq U(0)\), we obtain that \(\limsup_{t\rightarrow \infty } W(t)\leq \lim_{t\rightarrow \infty }U(t)=\frac{ \lambda }{n}\). This implies that the solution \((T(t),T_{s}(t),V_{s}(t),T _{r}(t), V_{r}(t))\) is also ultimately bounded. This completes the proof. □
Following the concept of the basic reproductive number for an epidemic disease presented in [16], we define the wildtype strain infection reproduction number \(R_{s}\) and the drugresistant strain infection reproduction number \(R_{r}\) as follows:
The fraction \(\frac{1}{c}\) gives the average lifespan of a virus for strain i (\(i=r,s\)). \(\frac{\lambda }{d}\) is the steadystate target cell density at the beginning of the strain i infection process (i.e. near the infectionfree steady state). \(k_{i}N_{i}\) gives the magnitude of virus particles produced by one strain i infectious (virusproducing) cell during its average survival time. Multiplying these quantities together gives the expected number of newly infected cells produced by a single newly for strain i infected cell, that is, \(R_{i}\).
Now, we discuss the equilibrium of model (2). The equilibrium can be given from the following equations:
Obviously, model (2) always has a unique infectionfree equilibrium \(E_{0}=(\frac{\lambda }{d},0,0,0,0)\). When \(T_{s}>0\) and \(T_{r}=0\), from (4) we directly have \(V_{r}=0\) and \(V_{s}T=0\), and then \(T_{s}=0\), which leads to a contradiction. When \(T_{s}=0\) and \(T_{r}>0\), from (4) we can obtain that if \(R_{r}>1\), then model (2) has a unique boundary equilibrium \(E_{r}=(T_{1},0,0,T_{r1},V_{r1})\) with
and if \(R_{r}\leq 1\), then \(E_{r}\) does not exist. When \(T_{s}>0\) and \(T_{r}>0\), from (4) we can obtain that
and
Clearly, functions \(T_{r}(T_{s})\) and \(T_{s}(T_{r})\) are decreasing in \(T_{s}\geq 0\) and \(T_{r}\geq 0\), respectively. We have
Furthermore, from \(T_{r}(T_{s})=0\) and \(T_{s}(T_{r})=0\), we obtain
It is easy to verify that \(T_{s}(0)< T_{s}^{*}\) when \((1u)R_{s}>1\). From \(T_{r}(0)< T_{r}^{*}\) we let
This shows that curves \(T_{r}(T_{s})\) and \(T_{s}(T_{r})\) have a unique intersection point \((T_{sc},T_{rc})\) in the positive quadrant, which means \(E_{c}=(T_{c},T_{sc},V_{sc},T_{rc},V_{rc})\) is the unique positive equilibrium of model (2). Thus, we finally have the following results.
Theorem 2

(i)
Model (2) always has a unique infectionfree equilibrium \(E_{0}\);

(ii)
If \(R_{r}>\max \{1,(1u)R_{s}+((1u)R_{s}1)\alpha _{1}\frac{ \lambda }{c}N_{r}\}\), then model (2) only has equilibria \(E_{0}\)and \(E_{r}\);

(iii)
If \((1u)R_{s}>1\geq R_{r}\), then model (2) only has equilibria \(E_{0}\)and \(E_{c}\);

(iv)
If \((1u)R_{s}+((1u)R_{s}1)\alpha _{1}\frac{\lambda }{c}N_{r}>R _{r}>1\), then model (2) has three equilibria \(E_{0}\), \(E_{r}\), and \(E_{c}\).
The existence of equilibria for model (2) is also intuitively expressed in Fig. 1. From Theorem 2 and Fig. 1, we can find that the saturated coefficient \(\omega _{1}\) of wildtype virus \(V_{s}\) has no effect on Fig. 1. Along with the decreasing of saturated coefficient \(\alpha _{1}\), the orange region will shrink, and it finally becomes the region \(\{(R_{s},R_{r}): (1u)R_{s}>R_{r}>1\}\) as \(\alpha _{1}\to 0\). On the contrary, along with the increasing of saturated coefficient \(\alpha _{1}\), the orange region will enlarge, and it finally becomes the region \(\{(R_{s},R_{r}): (1u)R_{s}>1, R_{r}>1\}\) as \(\alpha _{1} \to +\infty \).
Stability of equilibrium
Let \(E=(T,T_{s},V_{s},T_{r},V_{r})\) be any equilibrium of model (2). By calculating, we get that the Jacobian matrix at equilibria E is
Firstly, for the stability of equilibrium \(E_{0}\), we have the following results.
Theorem 3

(a)
If \((1u)R_{s}<1\)and \(R_{r}<1\), then infectionfree equilibrium \(E_{0}\)is locally asymptotically stable.

(b)
If \(R_{s}\leq 1\)and \(R_{r}\leq 1\), then \(E_{0}\)is globally asymptotically stable.

(c)
If \((1u)R_{s}>1\)or \(R_{r}>1\), then \(E_{0}\)is unstable.
Proof
At equilibrium \(E_{0}\), from (7) the characteristic equation of \(J(E_{0})\) is
One root of (8) is \(X_{1}=d<0\). When \((1u)R_{s}<1\) and \(R_{r}<1\), by the Routh–Hurwitz criterion, all roots of the equations
and
have negative real parts, respectively. This implies that \(E_{0}\) is locally asymptotically stable. When \((1u)R_{s}>1\) or \(R_{r}>1\), we easily see that equation (9) or (10) has at least a root with positive real part. This implies that \(E_{0}\) is unstable.
For the global stability of \(E_{0}\), we define Lyapunov function \(L_{1}(t)\) as follows:
We have
When \(R_{s}\leq 1\) and \(R_{r}\leq 1\), then \(\frac{\mathrm {d}L_{1}(t)}{\mathrm {d}t}\leq 0\) and the set \(M=\{(T,T_{s},V_{s},T _{r},V_{r}): \frac{\mathrm {d}L_{1}(t)}{\mathrm {d}t}=0\}\subset \{(T,T_{s},V _{s},T_{r},V_{r}): T=T_{0},T_{s}\geq 0,V_{s}\geq 0,T_{r}\geq 0,V_{r} \geq 0\}\).
For any solution trajectory \(\{(T(t),T_{s}(t),V_{s}(t),T_{r}(t),V_{r}(t)): t\geq 0\}\subset M\), we have \(T(t)\equiv T_{0}\). From the first equation of model (4), we obtain \(\frac{k_{s}V_{s}(t)T_{0}}{1+\omega _{1}V_{s}(t)}+\frac{k_{r}V_{r}(t)T _{0}}{1+\alpha _{1}V_{r}(t)}\equiv 0\), which implies \(V_{s}(t)=V_{r}(t) \equiv 0\). From the third and fifth equations of model (4), we also get \(N_{s}\delta T_{s}(t)cV_{s}(t)=0\) and \(N_{r}\delta T_{r}(t)cV_{r}(t)=0\), which further imply \(T_{s}(t)=T _{r}(t)\equiv 0\). Hence, \((T(t),T_{s}(t),V_{s}(t),T_{r}(t),V_{r}(t)) \equiv E_{0}\). From LaSalle’s invariance principle [17], \(E_{0}\) is globally asymptotically stable. This completes the proof. □
Remark 1
In Theorem 3, we only obtained the global asymptotic stability of \(E_{0}\) under \(R_{s}\leq 1\) and \(R_{r}\leq 1\). Therefore, based on conclusion (a) of Theorem 3, an interesting open problem is whether we can establish the global asymptotic stability of \(E_{0}\) when \((1u)R_{s}\leq 1\) and \(R_{r}\leq 1\).
Next, about the stability of equilibrium \(E_{r}\), we have the following results.
Theorem 4

(a)
If \(R_{r}>\max \{1,(1u)R_{s}+((1u)R_{s}1)\alpha _{1}\frac{ \lambda }{c}N_{r}\}\), then equilibrium \(E_{r}\)is locally asymptotically stable.

(b)
If \(R_{r}>1\)and \(R_{r}<(1u)R_{s}+((1u)R_{s}1)\alpha _{1}\frac{ \lambda }{c}N_{r}\), then \(E_{r}\)is unstable.

(c)
If \(R_{r}>\max \{1,R_{s}+\alpha _{1}\frac{\lambda }{c}N_{r}(R _{s}1)\}\), then \(E_{r}\)is globally asymptotically stable.
Proof
At equilibrium \(E_{r}\), from (7) the characteristic equation of \(J(E_{r})\) is
where
When \(R_{r}>\max \{1,(1u)R_{s}+((1u)R_{s}1)\alpha _{1}\frac{\lambda }{c}N_{r}\}\), we have \(a_{i}>0\) and \(b_{i}>0\) for \(i=0,1,2\). Since
According to the Routh–Hurwitz criterion, all roots of equation (11) have negative real parts. Therefore, \(E_{r}\) is locally asymptotically stable. When \(R_{r}>1\) and \(R_{r}<(1u)R_{s}+((1u)R _{s}1)\alpha _{1}\frac{\lambda }{c}N_{r}\), the equation \(X^{2}+a_{1}X+a _{0}=0\) has at least a positive real part root. This implies that \(E_{r}\) is unstable.
To obtain the global stability of \(E_{r}\), we define Lyapunov function \(L_{2}(t)\) as follows:
We have
Obviously, when \(R_{r}>\max \{1,R_{s}+\alpha _{1}\frac{\lambda }{c}N _{r}(R_{s}1)\}\), we have \(\frac{\mathrm {d}L_{2}(t)}{\mathrm {d}t}\leq 0\) and the set \(M=\{(T,T_{s},V_{s},T_{r},V_{r}): \frac{\mathrm {d}L_{2}(t)}{\mathrm {d}t}=0 \}\subseteq \{(T,T_{s},V_{s},T_{r},V_{r}): T=T_{1},T_{s}\geq 0,V_{s} \geq 0,T_{r}=T_{r1},V_{r}=V_{r1}\}\). From \(T(t)\equiv T_{1}\), \(T_{r}(t)\equiv T_{r1}\) and \(V_{r}(t)\equiv V_{r1}\), we have \(\lambda dT_{1}\frac{k_{s}V_{s}(t)T_{1}}{1+\omega _{1}V_{s}(t)}\frac{k _{r}V_{r1}T_{1}}{1+\alpha _{1}V_{r1}}\equiv 0\), which implies \(V_{s}(t)\equiv 0\). From the third equation of model (4), we get \(N_{s}\delta T_{s}(t)cV_{s}(t)\equiv 0\), which implies \(T_{s}(t) \equiv 0\). Hence, \((T(t),T_{s}(t),V_{s}(t),T_{r}(t),V_{r}(t))\equiv E _{r}\). Thus, LaSalle’s invariance principle implies that \(E_{r}\) is globally asymptotically stable. This completes the proof. □
Remark 2
In Theorem 4, we only obtained the global asymptotic stability of \(E_{r}\) when \(R_{r}>\max \{1,R_{s}+\alpha _{1}\frac{\lambda }{c}N_{r}(R _{s}1)\}\). Therefore, combining conclusion (a) of Theorem 4 an interesting open problem is whether we can establish the global asymptotic stability of \(E_{r}\) when \(R_{r}>\max \{1,(1u)R_{s}+((1u)R _{s}1)\alpha _{1}\frac{\lambda }{c}N_{r}\}\).
Remark 3
It is regretful that we here do not establish the corresponding criteria on the local and global stability for positive equilibrium \(E_{c}\) of model (2). The reasons are that the analysis of the characteristic equation of \(J(E_{c})\) is very complex, and the construction of a suitable Lyapunov function is also very difficult. However, in the next section we can establish the uniform persistence of model (2) when positive equilibrium \(E_{c}\) exists.
Uniform persistence
Theorem 5
If \((1u)R_{s}>1\geq R_{r}\)or \((1u)R_{s}+((1u)R_{s}1)\alpha _{1}\frac{ \lambda }{c}N_{r}>R_{r}>1\), then model (2) is uniformly persistent. That is, there exists a positive constantδsuch that, for any positive solution \((T(t),T_{s}(t),V_{s}(t),T_{r}(t),V_{r}(t))\)of model (2),
Proof
For any \(x_{0}=(T_{0},T_{s0},V_{s0},T_{r0},V_{r0})\in R^{5}_{+}\), let \(u(t,x_{0})=(T(t,x_{0}), T_{s}(t,x_{0}), V_{s}(t,x_{0}), T_{r}(t,x_{0}), V_{r}(t,x_{0}))\) be the solution of model (2) with the initial condition \(u(0,x_{0})=x_{0}\). From the proof of Theorem 1, we have \(\limsup_{t\rightarrow \infty }u(t,x_{0})\leq \frac{\lambda }{n}\), where \(n=\min \{d,\frac{\delta }{2},c\}\). Hence, for any constant \(\epsilon >0\), there is \(T_{0}>0\), when \(t\geq T_{0}\) we get \(u(t,x_{0})<\frac{\lambda }{n}+\epsilon \). Then, from the first equation of model (2), we have
From the comparison theorem and the arbitrariness of ϵ, we have
This shows that \(T(t,x_{0})\) is uniformly persistent.
Define
The boundary of X is
Denote
Let \(\omega (x_{0})\) be the ωlimit set of solution \(u(t,x_{0})\). Then we consider the following two cases.
Case (1): \((1u)R_{s}>1\geq R_{r}\). From Theorem 2, model (2) has only two equilibria \(E_{0}\) and \(E_{c}\). Let \(M_{0}=\{E_{0}\}\). It is clear that \(M_{0}\subset \bigcup_{x_{0}\in M_{\partial }}\omega (x _{0})\). For any \(x_{0}\in M_{\partial }\), let \(x_{0}=(T_{0},T_{s0},V _{s0},T_{r0},V_{r0})\). Due to \(u(t,x_{0})\in \partial X\) for all \(t\geq 0\), we have \(T_{s}(t,x_{0})\equiv 0\) or \(V_{s}(t,x_{0})\equiv 0\) or \(T_{r}(t,x_{0})\equiv 0\) or \(V_{r}(t,x_{0})\equiv 0\). If \(T_{s}(t,x_{0})\equiv 0\), then from the second equation of model (2), we have \(V_{s}(t,x_{0})\equiv 0\). Thus, model (2) degenerates into the following form:
If \(T_{r0}+V_{r0}=0\), then from system (12) we can obtain \(T_{r}(t,x _{0})\equiv V_{r}(t,x_{0})\equiv 0\). Thus, model (2) can further degenerate into
It follows that \(\lim_{t\rightarrow \infty }T(t,x_{0})=\frac{\lambda }{d}=T_{0}\). This shows that \(\omega (x_{0})=E_{0}\subset M_{0}\).
If \(T_{r0}+V_{r0}>0\), without loss of generality, we assume \(T_{r0}>0\) and \(V_{r0}\geq 0\). From the second equation of system (12), we can obtain \(T_{r}(t,x_{0})\geq T_{r0}e^{\delta t}>0\) for all \(t\geq 0\), and then, from the third equation of (12), we further obtain \(V_{r}(t,x_{0})>V_{r0}e^{ct}\geq 0\) for all \(t>0\). Choose a Lyapunov function as follows:
We obtain
and \(\{(T,T_{r},V_{r}): \frac{\mathrm {d}U_{0}(t)}{\mathrm {d}t}=0\}\subset \{(T,T _{r},V_{r}): T=T_{0}\}\). If \(T(t,x_{0})\equiv T_{0}\), then from the first equation of system (12), we have \(V_{r}(t,x_{0})\equiv 0\); further, from the third equation of system (12), we have \(T_{r}(t,x _{0})\equiv 0\). Thus, LaSalle’s invariance principle [17] implies that \((T(t,x_{0}),T_{r}(t,x_{0}),V_{r}(t,x_{0}))\rightarrow (T_{0},0,0)\) when \(t\rightarrow \infty \). This shows that \(\omega (x_{0})=E_{0}\subset M_{0}\).
If \(V_{s}(t,x_{0})\equiv 0\), from the third equation of model (2), we have \(T_{s}(t,x_{0})\equiv 0\). Similar to the above argument, we also get \(\omega (x_{0})=E_{0}\subset M_{0}\).
If \(T_{r}(t,x_{0})\equiv 0\), from the fourth equation of model (2), we have \(V_{s}(t,x_{0})\equiv 0\) and \(V_{r}(t,x_{0})\equiv 0\). Then, from the third equation of model (2), we have \(T_{s}(t,x_{0})\equiv 0\). Thus, model (2) degenerates into
It follows that \(\lim_{t\rightarrow \infty }T(t,x_{0})=T_{0}\). This shows that \(\omega (x_{0})=E_{0}\subset M_{0}\).
If \(V_{r}(t,x_{0})\equiv 0\), from the fifth equation of model (2), we get \(T_{r}(t,x_{0})\equiv 0\). Similar to the above argument, we also get \(\omega (x_{0})=E_{0}\subset M_{0}\).
Finally, we have \(M_{0}=\bigcup_{x_{0}\in M_{\partial }}\omega (x_{0})\). Furthermore, it is clear that \(M_{0}\) is isolated invariant and noncycle in ∂X.
Now, we prove that \(W^{s}(E_{0})\cap X=\emptyset \), where \(W^{s}(E _{0})\) is the stable set of \(E_{0}\). Suppose that there is an \(x_{0}\in X\) such that \(\lim_{t\rightarrow \infty }u(t,x_{0})=E_{0}\), then we have \(\lim_{t\rightarrow \infty }T(t,x_{0})=T_{0}\). Hence, for any constant \(\epsilon >0\), there is \(T^{*}>0\) such that \(T(t,x_{0}) \geq T_{0}\epsilon \) and \(V_{s}(t,x_{0})<\epsilon \) for any \(t\geq T^{*}\). Define the function
We have \(\lim_{t\rightarrow \infty }U_{1}(t,x_{0})=0\). When \(t\geq T^{*}\), we have
Due to \((1u)R_{s}>1\), we choose enough small \(\epsilon >0\) such that \((1u)\frac{k_{s}(T_{0}\epsilon )}{1+\omega _{1}\epsilon }\frac{c}{N _{s}}>0\). Thus, \(U_{1}(t)\) is increasing for \(t\geq T^{*}\). Hence, we know that \(U_{1}(t)\) does not tend to zero as \(t\to \infty \), which leads to a contradiction. This shows that \(W^{s}(E_{0})\cap X=\emptyset \). According to the theory of persistence in dynamical systems (see [18]), there is a constant \(\delta >0\) such that, for any \(x_{0} \in X\), one has
This shows that model (2) is uniformly persistent.
Case (2): \((1u)R_{s}+((1u)R_{s}1)\alpha _{1}\frac{\lambda }{c}N_{r}>R_{r}>1\). From Theorem 2, model (2) has three equilibria \(E_{0}\), \(E_{r}\), and \(E_{c}\). Denote \(M_{0}=\{E_{0},E_{r}\}\). It is clear that \(M_{0}\subset \bigcup_{x_{0}\in M_{\partial }}\omega (x_{0})\). For any \(x_{0}\in M_{\partial }\), let \(x_{0}=(T_{0},T_{s0},V_{s0},T _{r0},V_{r0})\). Due to \(u(t,x_{0})\in \partial X\) for all \(t\geq 0\), we have \(T_{s}(t,x_{0})\equiv 0\) or \(V_{s}(t,x_{0})\equiv 0\) or \(T_{r}(t,x_{0})\equiv 0\) or \(V_{r}(t,x_{0})\equiv 0\). If \(T_{s}(t,x _{0})\equiv 0\), then, similar to the above argument, model (2) degenerates into system (12).
If \(T_{r0}+V_{r0}=0\), from a similar argument as in case (1), we can obtain \(\omega (x_{0})=E_{0}\subset M_{0}\).
If \(T_{r0}+V_{r0}>0\), then we also can obtain \(T_{r}(t,x_{0})>0\) and \(V_{r}(t,x_{0})>0\) for all \(t>0\). Choose the Lyapunov function
Then we have
and the set \(\{(T,T_{r},V_{r}): \frac{\mathrm {d}U_{2}(t)}{\mathrm {d}t}=0\}=\{(T _{1},T_{r1},V_{r1})\}\). Hence, LaSalle’s invariance principle [17] implies that \((T(t,x_{0}),T_{r}(t,x_{0}),V_{r}(t,x_{0}))\rightarrow (T _{1},T_{r1},V_{r1})\) as \(t\rightarrow \infty \). This shows that \(\omega (x_{0})=E_{r}\subset M_{0}\).
If \(V_{s}(t,x_{0})\equiv 0\) or \(T_{r}(t,x_{0})\equiv 0\) or \(V_{r}(t,x _{0})\equiv 0\), then, following a similar argument as in case (1), we can also obtain \(\omega (x_{0})=E_{0}\) or \(\omega (x_{0})=E_{r}\), and hence \(\omega (x_{0})\subset M_{0}\).
Finally, we have \(M_{0}=\bigcup_{x_{0}\in M_{\partial }}\omega (x_{0})\). Furthermore, it is clear that \(E_{0}\) and \(E_{r}\) are isolated invariant and \(M_{0}\) is noncycle in ∂X.
Now, we prove that \(W^{s}(E_{0})\cap X=\emptyset \) and \(W^{s}(E_{r}) \cap X=\emptyset \). Similar to the above argument in case (1) we can get \(W^{s}(E_{0})\cap X=\emptyset \). Suppose that there is \(x_{0}\in X\) such that \(\lim_{t\rightarrow \infty }u(t,x_{0})=E_{r}\), then we have \(\lim_{t\rightarrow \infty }T(t,x_{0})=T_{1}\). Hence, for any constant \(\epsilon >0\), there is \(T^{*}>0\) such that \(T(t,x_{0})\geq T_{1} \epsilon \) and \(V_{s}(t,x_{0})<\epsilon \) for any \(t\geq T^{*}\). Define the function
We have \(\lim_{t\rightarrow \infty }U_{3}(t,x_{0})=0\). When \(t\geq T^{*}\), we have
Due to \((1u)R_{s}+((1u)R_{s}1)\alpha _{1}\frac{\lambda }{c}N_{r}>R _{r}>1\), we choose enough small \(\epsilon >0\) such that \((1u)\frac{k _{s}(T_{1}\epsilon )}{1+\omega _{1}\epsilon }\frac{c}{N_{s}}>0\). Then \(U_{3}(t)\) is increasing for \(t\geq T^{*}\). Thus, we know that \(U_{3}(t)\) does not tend to zero, which leads to a contradiction. Hence, \(W^{s}(E_{r})\cap X=\emptyset \). According to the theory of persistence in dynamical systems (see [18]), there is a constant \(\delta >0\) such that, for any \(x_{0}\in X\), one has
This shows that model (2) is also uniformly persistent. This completes the proof. □
Remark 4
An interesting open problem is whether the positive equilibrium \(E_{c}\) is also globally asymptotically stable when the conditions in Theorem 5 are satisfied.
Numerical examples
In this section, we provide the numerical examples to illustrate the global asymptotic stability of the equilibria for model (2), and Examples 1 and 2 can further verify Remarks 1 and 2, respectively.
Example 1
In model (2), we take the parameters \(\lambda =10^{5}\), \(d=0.1\), \(k_{s}=1.0\times 10^{8}\), \(k_{r}=1.0 \times 10^{8}\), \(u=0.6\), \(\delta =1\), \(N_{s}=2000\), \(N_{r}=900\), \(c=11\), \(\omega _{1}=10^{5}\), and \(\alpha _{1}=10^{4}\). By calculating, we have \(R_{s}\approx 1.8182>1\), \((1u)R_{s}\approx 0.7273<1\), and \(R_{r}\approx 0.8182<1\). Furthermore, we also have the infectionfree equilibrium \(E_{0}=(10^{6}, 0, 0, 0, 0)\). We give three different groups of initial values in Table 2.
The numerical simulations given in Fig. 2 illustrate that equilibrium \(E_{0}\) may be globally asymptotically stable. This shows that the open problem given in Remark 1 may be right.
Example 2
In model (2), we take the parameters \(\lambda =10^{5}\), \(d=0.005\), \(k_{s}=1.2\times 10^{9}\), \(k_{r}=1.0 \times 10^{8}\), \(u=0.6\), \(\delta =1\), \(N_{s}=2000\), \(N_{r}=250\), \(c=10\), \(\omega _{1}=10^{3}\), and \(\alpha _{1}=10^{7}\). By calculating, we have \(R_{r}=5\), \((1u)R_{s}+((1u)R_{s}1)\alpha _{1}\frac{\lambda }{c}N_{r}=2.15\) and \(R_{s}+(R_{s}1)\alpha _{1}\frac{\lambda }{c}N_{r}=5.75\). Hence, \(\max \{1,(1u)R_{s}+((1u)R_{s}1)\alpha _{1}\frac{\lambda }{c}N_{r} \}< R_{r}< R_{s}+(R_{s}1)\alpha _{1}\frac{\lambda }{c}N_{r}\). Furthermore, we also have the boundary equilibrium \(E_{r}=(4.76\times 10^{6}, 0, 0, 7.62\times 10^{4}, 1.90\times 10^{6})\). We give three different groups of initial values in Table 3.
The numerical simulations given in Fig. 3 illustrate that equilibrium \(E_{r}\) may be globally asymptotically stable. This shows that the open problem given in Remark 2 may be right.
Example 3
In model (2), we take the parameters \(\lambda =10^{5}\), \(d=0.005\), \(k_{s}=1.2\times 10^{9}\), \(k_{r}=4.0 \times 10^{10}\), \(u=3\times 10^{5}\), \(\delta =1\), \(N_{s}=2000\), \(N_{r}=1000\), \(c=10\), \(\omega _{1}=10^{8}\), and \(\alpha _{1}=10^{2}\). By calculating, we have \((1u)R_{s}\approx 4.80\), \(R_{r}=0.8\), and \((1u)R_{s}>1\geq R_{r}\), and model (2) has a coexistence equilibrium \(E_{c}\approx (4.80\times 10^{6}, 1.6\times 10^{6}, 1.520\times 10^{7}, 2.416, 241.577)\). We give three different groups of initial values in Table 4.
The numerical simulations given in Fig. 4 illustrate that equilibrium \(E_{c}\) may be globally asymptotically stable. This shows that the open problem given in Remark 4 may be right.
Example 4
In model (2), we take the parameters \(\lambda =10^{5}\), \(d=0.005\), \(k_{s}=1.2\times 10^{8}\), \(k_{r}=1.0 \times 10^{8}\), \(u=3\times 10^{5}\), \(\delta =1\), \(N_{s}=2000\), \(N_{r}=1000\), \(c=10\), \(\omega _{1}=10^{8}\), and \(\alpha _{1}=10^{8}\). By calculating, we have \((1u)R_{s}+((1u)R_{s}1)\alpha _{1}\frac{ \lambda }{c}N_{r}\approx 52.798\), \(R_{r}=20\), and \((1u)R_{s}+((1u)R _{s}1)\alpha _{1}\frac{\lambda }{c}N_{r}>R_{r}>1\), and model (2) has a coexistence equilibrium \(E_{c}\approx (4.979\times 10^{5}, 9.750 \times 10^{4}, 1.950\times 10^{7}, 5.826, 582.635)\). We give three different groups of initial values in Table 5.
The numerical simulations given in Fig. 5 illustrate that equilibrium \(E_{c}\) may be globally asymptotically stable. This shows that the open problem given in Remark 4 may be right.
Conclusion
In this paper, we study the global dynamics for a twostrain HIV infection model with saturated incidence which includes wildtype (i.e. drug sensitive) and drugresistant strains. The wildtype strain can mutate and become drugresistant during the process of reverse transcription. The main results are presented in Theorems 1–5. Concretely, the nonnegativity and boundedness of solutions are obtained in Theorem 1; the existence of wildtype strainfree equilibrium and coexistence equilibrium is also obtained in Theorem 2; Theorems 3 and 4 show the sufficient and necessary threshold conditions for the local and global asymptotic stability of infectionfree and wildtype strainfree equilibria; and the uniform persistence of HIV infection model is established in Theorem 5.
There are some problems waiting for further investigation. Firstly, Remarks 1 and 2 consider an interesting open problem is whether we can establish the global asymptotic stability of equilibria under the appropriate conditions. And it is meaningful to study more complex models (see [19]), for example, a twostrain infection model with delayed saturation incidence (see [20]) and general nonlinear incidence (see [15, 21]), etc. Furthermore, it is more reasonable to consider the dynamical behaviors of a virus infection model with spatial diffusion and agedependence (see [22–25]). We will leave these problems for future investigation.
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Acknowledgements
We would like to thank the anonymous referees for their helpful comments and the editor for his constructive suggestions, which greatly improved the presentation of this paper.
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This research is supported by the Natural Science Foundation of China (Grant No. 11771373, 11861065) and the Natural Science Foundation of Xinjiang Province of China (Grant No. 2016D03022).
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Correspondence to Zhidong Teng.
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Chen, W., Tuerxun, N. & Teng, Z. The global dynamics in a wildtype and drugresistant HIV infection model with saturated incidence. Adv Differ Equ 2020, 25 (2020) doi:10.1186/s1366202024972
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Keywords
 HIV virus infection model
 Wildtype and drugresistant virus
 Saturated incidence
 Basic reproduction number
 Stability and persistence