# Global Exponential Stability of a Delayed HIV Infection Model with a Nonlinear Incidence Rate

Authors: Zhiwen Long
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

Under the assumption that there is a time delay between the time target cells are contacted by the virus particles and the time the contacted cells become actively infected, we investigate the exponential stability of the noninfected equilibrium for a delayed HIV infection model with a nonlinear incidence rate. Compared with the global asymptotic stability analysis based on basic reproduction number, exponential stability analysis reveals the change range of various cells in different time periods.


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## Body

## 1. Introduction

As is well-known, acquired immune deficiency syndrome (AIDS) has received widespread attention since its discovery. Initial models used to gain an insight into HIV immunology relied on systems of ordinary differential equations (see, e.g., [1–6]). In general, an underlying assumption in such an ODE model for HIV infection is that infection of cells by virions is instantaneous. In fact, in the real world, there may be intracellular delays in the viral infection and replication, and immune response processes. Furthermore, delay-differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could affect the stability of the systems and may lead to some complex dynamic behaviors such as oscillation, chaos, and instability [7, 8]. Hence, time delays have been incorporated into HIV infection models by some authors (see [9–16] and the references cited therein).On the other hand, compared with bilinear incidence rate, nonlinear incidence rate usually has more complex properties. When the number of viruses and susceptible cells is large, the number of susceptible cells contacted by viruses per unit time is limited, so the nonlinear incidence rate should be adopted in this case [17]. Recently, under the assumption that there is a lag between the time target cells are contacted by the virus particles and the time the contacted cells become actively infected, Yuan et al. [18] proposed the following delay-differential system for HIV infection models with a nonlinear incidence rate:(1)x′t=μ−kxt−αxtfvt,y′t=αe−mτxt−τfvt−τ−γyt−βythzt,v′t=pyt−dvt,z′t=δyt−qzt,where xt,yt,vt, and zt denote CD4+ cells that are susceptible to infection, productively infected cells, virus, and the effector population of CTLs (cytotoxic T lymphocytes), respectively, at time t; μ denotes the newly added susceptible cell, k represents its death rate constant, α is the infection rate constant, γ signifies the infected cell death rate constant, and β represents the killing rate constant of productively infected cell by CTLs; p denotes the rate constant of virus production by infected cell, and d determines the clearance rate constant of virus; effectors are generated in the presence of infected cells at rate δy and die at rate constant q per cell; τ>0 represents lag between the time target cells are contacted by the virus particles and the time the contacted cells become actively infected, and e−mτ is assumed to describe the surviving rate constant of each target cell to get infected. The functions fx and hx represent the force of infection by the infective at density x and the force of CTLs to kill infected cells at density x, respectively. Furthermore, the functions fx and hx are locally Lipschitz on 0,∞ and satisfy the following:(A1)
f0=0, the derivatives f′x>0 and fx/x′≤0 in 0,∞(A2)
hx≥0 in 0,∞Due to their biological relevance, all parameters in model (1) are positive. In fact, the first equation of (1) has more general form x′t=nxt−gxt,vt, where nx is a general function that accounts for both production and turnover of healthy target cells [19, 20]. Generally speaking, there are two main forms of this function. In addition to nxt=λ−dxt [21, 22], another typical function appearing in the literature is nxt=λ−dxt+rxt1−xt/K, where λ,d,r,K are positive real numbers [23, 24]. In particular, if we allow r=0, the second form of nxt will become the first one. In order to include the aforementioned two forms, we consider the following delay-differential system:(2)x′t=μ−kxt+ρxt1−xtM−αxtfvt,y′t=αe−mτxt−τfvt−τ−γyt−βythzt,v′t=pyt−dvt,z′t=δyt−qzt.Here,0≤ρ<k, ρ is the maximum proliferation rate of uninfected cells, and M is the maximum level of uninfected cell concentration in the body. The other parameters are positive and have similar meanings to those in system (1).For the sake of convenience, we denote byℝn the set of all n-dimensional real vectors. For any x=x1,x2,…,xn∈ℝn, we let x denote the absolute-value vector given by x=x1,x2,…,xn and define x=maxi∈1,2,…,nxi. Let ℝ+ denote nonnegative real number space, C=C−τ,0,ℝ be the Banach space of continuous functions mapping the interval −τ,0 into ℝ equipped with the usual supremum norm ·, and let C+=C−τ,0,ℝ+. Set xtθ=xt+θ for all θ∈−τ,0.From the biological meanings, the initial conditions associated with (2) are defined as follows:(3)xθ=φθ,vθ=ψθ,y0,z0∈ℝ+,θ∈−τ,0,where φ,ψ∈C−τ,0,ℝ+.Most recently, by using the characteristic equation and the Fluctuation lemma, Yuan et al. [18] proved the following result:Theorem 1.
Let the basic reproduction number(4)R0=αpμe−mτf′0kγd<1.
Then, the noninfected equilibriumE0=μ/k,0,0,0 of (1) is globally asymptotically stable.However, Theorem1 does not give us any information about the convergence rate, which is vitally important to the disease prevention and control in real-world applications of theoretical results on epidemic models. In fact, the known convergence rate means that the range of the population is predictable; that is to say, we can estimate the range of changes in the population within a given time range. In particular, since the exponential convergence rate reveals the variation range of population in different time periods, there have been extensive results on the problem of the exponential stability of epidemic models in the literature studies [25–28]. Now, a question naturally arises: under what conditions is the noninfected equilibrium x∗,y∗,v∗,z∗=x∗ρ,0,0,0 of system (2) with initial conditions (3) are exponentially stable? Here,(5)x∗ρ=Mρ−k+M2ρ−k2+4ρMμ1/22ρ,ρ>0;x∗0=μk,ρ=0.It is easy to see thatlimρ⟶0+x∗ρ=x∗0=μ/k. For 0≤ρ<k, we denote x∗ρ by x∗ for simplicity of notation.Motivated by the above discussions, the main purpose of this paper is to establish sufficient conditions for the exponential stability of the noninfected equilibriumx∗,y∗,v∗,z∗=x∗,0,0,0 of system (2). To the best of our knowledge, it is the first time to focus on the problem of the exponential stability for (2). In particular, a numerical example is provided to illustrate our theoretical results.Similar to the reference [29], we give the definition of the exponential stability as follows.Definition 1.
Letxt,yt,vt,zt be the solution of (2) with initial value conditions (3). If there exists a positive constant λ>0 such that(6)xt−x∗=Oe−λt,yt−y∗=Oe−λt,vt−v∗=Oe−λt,zt−z∗=Oe−λt,as t⟶+∞, then the noninfected equilibrium is said to be globally exponentially stable.
In the rest of the paper, we give our main result in Section2. The theoretical result is illustrated with examples as well as numerical simulations in Section 3. Finally, conclusions are made in Section 4.


## 2. Main Results

By the fundamental theory of functional differential equations [30], we have that there is a unique solution xt,yt,vt,zt satisfying system (2) with initial conditions (3). Furthermore, by a similar argument as that in Zhu et al. [31], it is not hard to show that every solution of (2) with initial value conditions (3) is nonnegative and bounded on 0,∞. From (2), we have(7)x′t≤μ−k−ρxt,which implies(8)xt≤x0e−k−ρt+μ1−e−k−ρtk−ρ=μk−ρ+x0−μk−ρe−k−ρt,for all t∈0,∞.In addition, from (A1), we can get(9)fξ≤f′0ξ,ξ∈0,∞.The following theorem is our main result.Theorem 2.
Assume that the following(10)αf′0<mink−ρ+ρx∗/M−ρμ/Mk−ρx∗,γemτk−ρμ,p<d,δ<q,hold. Then, there exist three positive constants λ, K, and tξ such that(11)xt−x∗≤Ke−λt,yt−y∗≤Ke−λt,vt−v∗≤Ke−λt,zt−z∗≤Ke−λt,for all t>tξ. Here, x∗,y∗,v∗,z∗=x∗ρ,0,0,0, and(12)x∗ρ=Mρ−k+M2ρ−k2+4ρMμ1/22ρ,ρ>0;x∗0=μk,ρ=0.Proof.
Let(13)Xt=x1t,x2t,x3t,x4t=xt−x∗,yt−0,vt−0,zt−0,then we can get(14)x1′t=−kx1t+ρx1t−ρMx∗x1t−ρMxtx1t−αx1tfx3t−αx∗fx3t,x2′t=αe−mτxt−τfx3t−τ−γx2t−βx2thx4t,x3′t=px2t−dx3t,x4′t=δx2t−qx4t,x1t=e−∫t0tk−ρ+ρMx∗+αfx3sdsx1t0+∫t0te−∫vtk−ρ+ρMx∗+αfx3sds−ρMxvx1v−αx∗fx3vdv,x2t=e−∫t0tγ+βhx4sdsx2t0+∫t0te−∫vtγ+βhx4sdsαe−mτxv−τfx3v−τdv,x3t=e−dt−t0x3t0+∫t0te−dt−vpx2vdv,x4t=e−qt−t0x4t0+∫t0te−qt−vδx2vdv,for all t≥t0.
From (8) and (10), there exist t∗>0 and ε0>0 such that(15)xt≤ε0+μk−ρ,forallt∈t∗,∞,αe−mτf′0μk−ρ+ε0<γ.
Then, for anyε∈0,ε0, we can choose two positive constants λ and η such that(16)λ+ρMμk−ρ+ε+αx∗f′0−k−ρ+ρMx∗<−η<0,λ+αeλτe−mτf′0μk−ρ+ε−γ<−η<0,λ+p−d<−η<0,λ+δ−q<−η<0.
For a givenε∈0,ε0, from (8), we can find t1>0 such that(17)xt≤ε+μk−ρ,forallt∈t1,∞.
Lettξ=t1+τ,Kφ>1, and(18)Xξ=maxmaxs∈t0−τ,tξx1s,maxs∈t0−τ,tξx2s,maxs∈t0−τ,tξx3s,maxs∈t0−τ,tξx4s.
Consequently,(19)Xtξ<Xξ+ε<KφXξ+ε=KφXξ+εeλtξe−λtξ.
In the following, we will show(20)Xt<KφXξ+εeλtξe−λt,forallt>tξ.
If not, one of the following four cases must occur.Case I: there existsθ1>0 such that(21)x1θ1=KφXξ+εeλtξe−λθ1,Xt<KφXξ+εeλtξe−λt,for allt∈t0−τ,θ1.Case II: there existsθ2>0 such that(22)x2θ2=KφXξ+εeλtξe−λθ2,Xt<KφXξ+εeλtξe−λt,for allt∈t0−τ,θ2.Case III: there existsθ3>0 such that(23)x3θ3=KφXξ+εeλtξe−λθ3,Xt<KφXξ+εeλtξe−λt,for allt∈t0−τ,θ3.Case IV: there existsθ4>0 such that(24)x4θ4=KφXξ+εeλtξe−λθ4,Xt<KφXξ+εeλtξe−λt,for allt∈t0−τ,θ4.
If Case I holds, in view of (9), (14), (16), (17) and (21), we have(25)x1θ1=e−∫tξθ1k−ρ+ρMx∗+αfx3sdsx1tξ+∫tξθ1e−∫vθ1k−ρ+ρMx∗+αfx3sds−ρMxvx1v−αx∗fx3vdv≤e−k−ρ+ρMx∗θ1−tξx1tξ+∫tξθ1e−k−ρ+ρMx∗θ1−vρMμk−ρ+εx1v+αx∗f′0x3vdv≤e−k−ρ+ρMx∗θ1−tξXξ+ε+∫tξθ1e−k−ρ+ρMx∗θ1−vρMμk−ρ+ε+αx∗f′0KφXξ+εeλtξe−λvdv≤KφXξ+εeλtξe−λθ11Kφe−θ1−tξk−ρ+ρMx∗−λ+∫tξθ1e−θ1−vk−ρ+ρMx∗−λρMμk−ρ+ε+αx∗f′0dv≤KφXξ+εeλtξe−λθ11Kφe−θ1−tξk−ρ+ρMx∗−λ+∫tξθ1e−θ1−vk−ρ+ρMx∗−λk−ρ+ρMx∗−λdv=KφXξ+εeλtξe−λθ11−1−1Kφe−θ1−tξk−ρ+ρMx∗−λ<KφXξ+εeλtξe−λθ1,which contradicts with the first equation in (21). Hence, Case I could not hold.
If Case II holds, combining (A2), (9), (14), (16) and (17), and (22), we deduce that(26)x2θ2=e−∫tξθ2γ+βhx4sdsx2tξ+∫tξθ2e−∫vθ2γ+βhx4sdsαe−mτxv−τfx3v−τdv≤e−θ2−tξγx2tξ+∫tξθ2e−θ2−vγαe−mτf′0μk−ρ+εx3v−τdv=e−θ2−tξγXξ+ε+∫tξθ2e−θ2−vγαe−mτf′0μk−ρ+εKφXξ+εeλtξe−λv−τdv≤KφXξ+εeλtξe−λθ21Kφe−θ2−tξγ−λ+∫tξθ2e−θ2−vγ−λγ−λdv=KφXξ+εeλtξe−λθ21−1−1Kφe−θ2−tξγ−λ<KφXξ+εeλtξe−λθ2,which contradicts the first equation in (22). Thus, (22) could not hold.
If Case III holds, together with (14) and (16), (23) yields(27)x3θ3=e−dθ3−tξx3tξ+∫tξθ3px2se−dθ3−sds≤e−dθ3−tξXξ+ε+∫tξθ3pe−dθ3−sKφXξ+εeλtξe−λsds≤KφXξ+εeλtξe−λθ31Kφe−θ3−tξd−λ+∫tξθ3e−θ3−sd−λpds≤KφXξ+εeλtξe−λθ31Kφe−θ3−tξd−λ+∫tξθ3e−θ3−sd−λd−λds=KφXξ+εeλtξe−λθ31−1−1Kφe−θ3−tξd−λ<KφXξ+εeλtξe−λθ3,which contradicts the first equation in (23). Thus, (23) could not hold.
If Case IV holds, together with (14) and (16), (24) yields (28)x4θ4=|e−qθ4−tξx4tξ+∫tξθ4δx2se−qθ4−sds≤e−qθ4−tξXξ+ε+∫tξθ4e−qθ4−sδKφXξ+εeλtξe−λsds≤KφXξ+εeλtξe−λθ41Kφe−θ4−tξq−λ+∫tξθ4e−θ4−sq−λδds≤KφXξ+εeλtξe−λθ41Kφe−θ4−tξq−λ+∫tξθ4e−θ4−sq−λq−λds=KφXξ+εeλtξe−λθ41−1−1Kφe−θ4−tξq−λ<KφXξ+εeλtξe−λθ4,which contradicts with the first equation in (24). Thus, (24) could not hold.
Therefore, we obtain from the above discussions that(29)Xt<KφXξ+εeλtξe−λt,for allt>tξ,holds. Letting ε⟶0+, we have that(30)Xt≤KφXξeλtξe−λt,for allt>tξ,which proves (11), where K=KφXξeλtξ. This completes the proof of Theorem 2.
According to the above discussion, it is easy to see that our result is also true whenρ=0. Furthermore, compared with Theorem 1, we find that the significantly stronger conclusion of Theorem 2 is obtained with only slightly stricter conditions.


## 3. A Numerical Example

In this section, we will show the existence and global exponential stability of the noninfected equilibrium of system (2) by a numerical example.Letμ=24,k=0.3,ρ=0.05,M=80,α=0.01,m=2,τ=1,γ=0.25,β=0.03,p=0.02,d=0.05,δ=0.04,q=0.07, fx=0.2x,andhx=x+0.5sinx. Then,(31)x∗ρ=Mρ−k+M2ρ−k2+4ρMμ1/22ρ,ρ>0;x∗0=80,αf′0=0.002<0.003=mink−ρ+ρx∗/M−ρμ/Mk−ρx∗,γemτk−ρμ,p=0.02<d=0.05,δ=0.04<q=0.07,x′t=24−0.3xt+0.05xt1−xt80−0.002xtvt,y′t=0.002e−2xt−1vt−1−0.25yt−0.03ytzt+0.5sinzt,v′t=0.02yt−0.05vt,z′t=0.04yt−0.07zt,which is globally exponentially stable, and all solutions of system (31) converge exponentially to the noninfected equilibrium 80,0,0,0 with the exponential convergent rate ζ≈0.002. This fact is verified by the numerical simulation in Figure 1.Figure 1
(a) The numerical solutionxt,yt,vt,zt of equation (31) for the initial values xs≡60,y0=40,vs≡30,z0=20,s∈−1,0; (b) the numerical simulation of logarithmic solution xt,yt,vt,zt of equation (31).
(a)(b)Since the results in [3, 4, 19, 32, 33] give no opinions about the global exponential stability of the HIV infection models, it is clear that all the results in the above references cannot be applicable to prove the global exponential stability of system (31).


## 4. Conclusion

We have proved the global exponential stability of the noninfected equilibrium for a delayed HIV infection model with a nonlinear incidence rate. It is worth pointing out that the required conditions are simple and easy to verify. It is natural to ask whether our methods in this paper are available to study the global exponential stability of the infected equilibrium of the delayed HIV infection models. It is an issue worth our further study. In addition, as pointed out in the literature [34, 35], the further extension of the model is to consider the case with reaction-diffusion term, which is also the focus of our further research.



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