# Threshold Dynamics for a Time-Periodic Viral Infection Model with Cell-to-Cell Transmission and Drug Treatments

Authors: Jinhu Xu; Guokun Huang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this study, a time-periodic viral infection model incorporating cell-to-cell infection and antiretroviral therapy has been investigated. The basic reproduction numberℛ0 has been defined as a threshold parameter which governs whether or not the disease dies out. Theoretical results indicate that the disease goes to extinction if ℛ0<1 and otherwise the disease will uniformly persist. The global stabilities of the equilibria for the corresponding autonomous model have been investigated by constructing suitable Lyapunov functions. Moreover, numerical simulations have been carried out to validate the obtained results. The results show that cell-to-cell infection mode may be a barrier to curing the viral infection and increasing the efficacy of protease inhibitors for blocking cell-to-cell infection which will benefit to weaken the severity of the viral infection.


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## Body

## 1. Introduction

Recently, much attention and great effort have been paid on modelling of HIV, and many models have been proposed and studied on HIV spreading. Many earlier models of HIV infection models describe the interaction between virus and target cells by assuming that the infected cells produce virions instantaneously [1, 2]. However, research studies have been carried out to show that a latent period exists before the infected cells are activated to produce virus [3–5]. Therefore, it is reasonable to introduce the latent period into a model. As we know, antiretroviral drugs can effectively suppress viral replication to a low level, but cannot eradicate the virus permanently. An important reason is that HIV provirus can reside in latently infected CD4 + T-cells, which can live longer and cannot be affected by antiretroviral drugs or immune responses, but can be activated to produce virus by relevant antigens [5]. Thereafter, motivated by this factor, many viral infection models with latent cells have been proposed and studied to describe this phenomena [6–13] and references therein. For example, Pankavich [6] proposed and studied the following viral infection model:(1)T′t=λ−dTT−k1−ηrtTV,L′t=p1−ηrtkTV−α+dLL,I′t=1−p1−ηrtkTV+αL−dII,VI′t=1−ηpNdII−dVVIt,VNI′t=ηpNdII−dVVNIt,where T,L,I,VI, and VNI represent the concentration of uninfected target T-cells, latent cells, productively infected T-cells, infectious virons, and noninfectious virons at time t, respectively. λ, k, α, and p are the production rate of T-cells, virus-to-cell infection rate, the activation rate of latent cells, and the fraction of infections leading to latency. dT,dL,dI,dV are the death rate of susceptible T-cells, latent cells, actively infected T-cells, and virions, respectively. N denotes the burst rate of actively infected cells. ηrt and ηp are the efficacies of RTIs (reverse transcriptase inhibitors) and PIs (protease inhibitors), respectively. The global dynamics of model (1) have been investigated in [6].Notice that the drug efficacy in model (1) is assumed to be a constant coefficient. In fact, drugs are often administered for patients periodically. As we known, drug concentration will reach a peak value within a very short time when a dose is administrated, and the concentration down to a lower value as time goes and then reaches a peak value again when another dose is administrated [14–16]. Therefore, drug concentration may vary periodically during the dose interval. Moreover, only cell-free infection has been considered in earlier work; the cell-to-cell transmission was not considered in model (1). However, a recent research work has shown that cell-to-cell transmission may be one of the main infection mode which leads to a failed therapy and potentially contribute to viral persistence [17]. Because a better understanding of the viral dynamics is very significant in terms of applications, thus motivated by these arguments many viral infection models with cell-to-cell transmission have been proposed and studied [18–29] and references therein. Besides, drugs’ efficacy about cell-to-cell infection was not taken into consideration in model (1). However, the results obtained in [30] show that PIs can effectively block cell-to-cell spread of HIV by preventing cleavage of viral polyproteins into functional subunits leading to the formation of immature noninfectious virus particles, while RTIs are less effective inhibitors of HIV cell-to-cell spread compared to virus-to-cell infection. To the best of our knowledge, the time-periodic viral infection model with cell-to-cell infection and latency have not been studied. Hence, motivated by the abovementioned work and arguments, we consider the following time-periodic model with two infection modes:(2)T′t=λ−dTTt−1−ηrttk1TVI1+m1VI−1−ηp1tk2TI1+m2I,L′t=p1−ηrttk1TVI1+m1VI+p1−ηp1tk2TI1+m2I−α+dLL,I′t=1−p1−ηrttk1TVI1+m1VI+1−p1−ηp1tk2TI1+m2I+αL−dII,VI′t=1−ηp2tNdII−dVVIt,VNI′t=ηp2tNdII−dVVNIt,where k2 is the infection rate of productively infected T-cells. Assume ηrtt,ηp1t,ηp2t:ℝ⟶0,1 are the efficiencies of RTIs and PIs, and we assume they are continuous and periodic in time t with a same period ω. Here, we considered two saturated incidence rates, where m1 and m2 are the saturation parameters and are positive constants. Other parameters have the same meaning of model (1). For convenience, we denote β1t=k11−ηrtt, β2t=k21−ηp1t, and at=1−ηp2t. Since the last equation of model (2) is independent with the others. Thus, we will focus on the following reduced model:(3)T′t=λ−dTTt−β1tTVI1+m1VI−β2tTI1+m2I,L′t=pβ1tTVI1+m1VI+pβ2tTI1+m2I−α+dLL,I′t=1−pβ1tTVI1+m1VI+1−pβ2tTI1+m2I+αL−dII,VI′t=NdIatI−dVVIt.This study is organized as follows. In Section2, preliminary results and the definition of the basic reproduction number are studied. In Section 3, global extinction of the disease and the uniform persistence are investigated in terms of the basic reproduction number. The global asymptotic stability of the infection equilibrium to the corresponding autonomous model are discussed by applying the method of Lyapunov functions. In Section 4, some numerical simulations are carried out. A brief conclusion and discussion ends the paper.


## 2. The Basic Reproduction Number

In this section, we investigate the definition of the basic reproduction numberℛ0 for model (3) according to the work [31, 32]. The following result shows that solutions of model (3) are bounded.Theorem 1.
The solutionsTt,Lt,It,VIt of model (3) are uniformly and ultimately bounded, i.e., there exist an M1>0 and t∗>0 such that Tt,Lt,It,VIt≤M,M,M,M, for t≥t∗.Proof.
From model (3), we can obtain that(4)dT+L+Idt=λ−dTT−dLL−dII≤λ−σT+L+I,where σ=mindT,dL,DI. Thus, there exists t1>0 such that T+L+I≤λ/σ, for t≥t1. It follows from the last equation of model (3); we have, for t≥t1, dVI/dt≤NdI·maxt∈0,ωat·λ/σ−dVVI, which implies that there exist t∗≥t1 such that VIt≤NdI·maxt∈0,ωat·λ/σ1/dV, for t≥t∗. Let M=maxλ/σ,NdI·maxt∈0,ωat·λ/σ1/dV. It then follows that Tt,Lt,It,VIt≤M,M,M,M, for t≥t∗. Hence, the solutions of model (3) are uniformly and ultimately bounded. This finishes the proof.
Letℝn,ℝ+n be the standard ordered n-dimensional Euclidean space with a norm ·. For u,v∈ℝn, we write u≥v if u−v∈ℝ+n, u>v if u−v∈ℝ+n/0, and u≫v if u−v∈Intℝ+n. Let At be a continuous, cooperative, irreducible, and ω-periodic n×n matrix function and ΦAt be the fundamental solution matrix of the following linear system:(5)dxdt=Atx.
LetrΦAω be the spectral radius of ΦAω. It follows from the Perron–Frobenius theorem that rΦAω is the principal eigenvalue of ΦAω in the sense that it is simple and admits an eigenvector v∗≫0. The following lemma comes from [33] which will be used for the discussion in the next section.Lemma 1 (see [33]).
Letμ=1/ωlnrΦAω. Then, there exists a positive ω -periodic function vt such that eμtvt is a solution of (5).
Obviously, model (3) has a unique infection-free equilibrium E0=T0,0,0,0, where T0=λ/dT. Linearizing (3) at E0 yields that(6)L′t=pβ2tT0I+pβ1tT0VI−α+dLL,I′t=1−pβ2tT0I+1−pβ1tT0VI+αL−dII,VI′t=NdIatI−dVVIt.
Define(7)F=0pβ2tT0pβ1tT001−pβ2tT01−pβ1tT0000,V=α+dL00−αdI00−NdIatdV.
Then, system (6) can be written as(8)dxdt=Ft−Vtxt.
Assume thatYt,s, t≥s, is the evolution operator of the following system:(9)dydt=−Vty.
Then, the3×3 matrix Yt,s satisfies(10)dYt,sdt=−VtYt,s,∀t≥s,Ys,s=Iis a3×3identity matrix.
LetCω be the ordered Banach space of all ω-periodic functions from ℝ to ℝ3, which is equipped with the maximum norm · and the positive cone Cω+≔ϕ∈Cω:ϕt≥0,∀t∈ℝ. Suppose ϕs∈Cω is the initial distribution of infectious cells and virus in this periodic environment; then, Fsϕs is the rate of new infections produced by the infected cells and virus who were introduced at time s, and Yt,sFsϕs represents the distribution of those infected cells and virus who were newly infected at time s and remain in the infected compartments at time t, for t≥s. Hence,(11)ψt≔∫−∞tYt,sFsϕsds=∫0∞Yt,t−aFt−aϕt−adais the distribution of accumulative new infections at time t produced by all those infected cells and virus introduced before t.
Define the linear operatorℒ:Cω⟶Cω as follows:(12)ℒϕt=∫0∞Yt,t−aFt−aϕt−ada,∀t∈ℝ,ϕ∈Cω.
It follows from the idea in [32] that the basic reproduction number ℛ0 of system (3) is defined as the spectral radius of ℒ, i.e.,(13)ℛ0=rℒ.
Moreover, the local asymptotic stability of the infection-free equilibriumE0 follows from [32].Theorem 2 (see [32]).
The following statements are valid:(i)
ℛ0<1 if and only if rΦF−Vω<1(ii)
ℛ0=1 if and only if rΦF−Vω=1(iii)
ℛ0>1 if and only if rΦF−Vω>1
Thus, infection-free equilibriumE0 of (3) is asymptotically stable if ℛ0<1 and unstable if ℛ0>1.


## 3. The Threshold Dynamics

### 3.1. Stability and Persistence of the Disease

In this section, we will investigate the global asymptotic stability of infection-free equilibrium and the disease persistence by regardingℛ0 as a threshold parameter.Theorem 3.
Ifℛ0<1, then the infection-free equilibrium E0 is globally asymptotically stable, and it is unstable for ℛ0>1.Proof.
It follows from Theorem2 that if ℛ0<1, then E0 is locally asymptotically stable and E0 is unstable when ℛ0>1. Hence, it is sufficient to show that E0 is global attractive for ℛ0<1.
From the first equation of (3) and nonnegativity of the solutions, we then have dT/dt≤λ−dTT, which implies that ∀ε>0; there exists t˜>0 such that Tt≤T0+ε, t>t˜.
Consider the following auxiliary system:(14)L˜′t=pβ2tT0+εI˜+pβ1tT0+εV˜I−α+dLL˜,I˜′t=1−pβ2tT0+εI˜+1−pβ1tT0+εV˜I+αL˜−dII˜,V˜I′t=NdIatI˜−dVV˜It,which is equivalent to(15)L˜′I˜′V˜I′=F−VL˜I˜V˜I+εMtL˜I˜V˜I,where(16)Mt=0pβ2tpβ1t01−pβ2t1−pβ1t000.
It then follows from Lemma1 that there exists a positive ω-periodic function vt=v1t,v2t,v3t such that eμtvt is a solution of (14), where μ=1/ωlnrΦF−V+εMω. Choose t¯>t˜ and a real number α1>0 such that L˜t¯,I˜t¯,V˜It¯T≤α1v0, which implies that(17)L˜t¯,I˜t¯,V˜It¯T≤α1eμt−t¯vt−t¯,t≥t¯.
The comparison principle yields that(18)Lt,It,VItT≤α1eμt−t¯vt−t¯,t≥t¯.
Recall Theorem2 that ℛ0<1 if and only if rΦF−Vω<1. Since the continuity of the spectrum for matrices [34], then choose ε>0 small enough such that rΦF−V+εMω<1, which implies that μ<0. Then, we have Lt,It,VItT⟶0 as t⟶∞. Furthermore, it follows from the first equation of model (3) and the theory of asymptotically periodic semiflows [35] that limt⟶∞Tt=T0. Thus, E0 is globally attractive.Theorem 4.
Ifℛ0>1, then there exists an ε>0 such that any solution Tt,Lt,It,VIt of model (3) with initial values T0,L0,I0,VI0=T0,L0,I0,VI0∈ℝ+×Intℝ+3; the solution of (3) satisfies liminft⟶∞Tt,Lt,It,VIt≥ε,ε,ε,ε and admits at least one positive periodic solution.Proof.
Let(19)X=ℝ+4,X0=ℝ+×Intℝ+3,∂X0=XX0.
Define Poincaree´ map P:ℝ+4⟶ℝ+4, satisfying Px0=uω,x0, ∀x0∈ℝ+4, with ut,x0 as the unique solution of (3) satisfying u0,x0=x0.
We first show thatP is uniformly persistent with respect to X0,∂X0. It is easy to see that X and X0 are positively invariant. Moreover, ∂X is a relatively closed set in X. Recall Theorem 1 that the solutions of model (3) are uniformly and ultimately bounded; thus, the semiflow P is point dissipative on ℝ+4, and P:ℝ+4⟶ℝ+4 is compact. Consequently, it follows from [36] that the semiflow P admits a global attractor, which attracts every bounded set in ℝ+4.
Define(20)M∂=T0,L0,I0,VI0∈∂X0:PmT0,L0,I0,VI0∈∂X0,∀m≥0.
Then, we claim thatM∂=T,0,0,0:T≥0. In fact, it is obvious that T,0,0,0:T≥0⊆M∂. For any T0,L0,I0,VI0∈∂X0/T,0,0,0:T≥0, considering the following cases: (1) L0=0,I0>0,VI0>0, (2) L0>0,I0=0,VI0>0, (3) L0>0,I0>0,VI0=0, (4) L0=I0=0,V0>0, (5) L0=VI0=0,I0>0, and (6) L0>0,I0=VI0=0. For case (1), we have dL/dt|t=0=pβ10T0VI0+pβ20T0I0>0, which implies that Tt,Lt,It,VIt∉∂X0, for 0<t≪1; then, Tt,Lt,It,VIt∉M∂. Similarly, for the other cases, it also has the same result; here, we omit the proof. Thus, for any T0,L0,I0,VI0∉T,0,0,0:T≥0, T0,L0,I0,VI0∉M∂, it indicates that M∂⊆T,0,0,0:T≥0.
Clearly,E0 is one fixed point of P in M∂. If Tt,Lt,It,VIt is a solution of model (3) from M∂, it then follows from that model (3) that Tt⟶T0,Lt⟶0,It⟶0,VIt⟶0 as t⟶∞.
Next, we will show that if the invariant setE0 is isolated, then E0 is an acyclic covering. To do this, it needs to prove any solution of model (3) initiating from M∂ will remain into M∂, which can be obtained easily. The isolated invariance of E0 will follow proof.
Now, we need to prove thatWsE0∩X0=∅. Denote x0=T0,L0,I0,VI0∈X0. Since the continuity of solutions with respect to the initial values, thus for ∀ε>0, there exists δ0>0 such that, for all x0∈X0 with x0−E0≤δ0, yields(21)ut,x0−ut,E0≤ε,∀t∈0,ω.
Then, we claim that(22)limsupm⟶∞dPmx0,E0≥δ0.
If it is not true, then we have(23)limsupm⟶∞dPmx0,E0<δ0.
For somex0∈X0, without loss of generality, we suppose that limsup/limits_t⟶∞dPmx0,E0<δ0, ∀m>0. Then, we can obtain that(24)ut,Pmx0−ut,E0∥≤ε,∀t∈0,ω.
For anyt≥0, let t=mω+t1, where t1∈0,ω and m=t/ω, which is the greatest integer less than or equal to t/ω. Then, we have(25)ut,Pmx0−ut,E0=ut1,Pmx0−ut1,E0≤ε,∀t∈0,ω.
SetTt,Lt,It,VIt=ut,x0; then, we have T0−ε≤T≤T0+ε, 0≤L≤ε, 0≤I≤ε, and 0≤VI≤ε, for t≥0. Then, T/1+m1VI≥T0−ε/1+m1ε=T0−ε1+m1T0/1+m1ε≥T0−ε1+m1T0 and T/1+m2I≥T0−ε/1+m2ε=T0−ε1+m2T0/1+m2ε≥T0−ε1+m2T0. It follows from model (3) that(26)L′t≥pβ1tT0−ε1+m1T0VI+pβ2tT0−ε1+m2T0I−α+dLL,I′t≥1−pβ1tT0−ε1+m1T0VI+β2tT0−ε1+m2T0I+αL−dII,VI′t=NdIatI−dVVIt.
Set(27)Mε=0ε1+m2T0pβ2tε1+m1T0pβ1t0ε1+m2T01−pβ2tε1+m1T01−pβ1t000.
It follows from Theorem2 that rΦF−Vω>1; then, we can select ε>0 small enough such that rΦF−V−Mεω>1. It follows from Lemma 1 and the standard comparison principle that there exists a positive ω-periodic function v¯t=v¯1t,v¯2t,v¯3tT such that Qt≥v¯teμ1t, where Qt=Lt,It,VItT and μ1=1/ωlnrΦF−V−Mεω>0, which implies that lim/limits_t⟶∞Lt,It,VIt=∞; this is a contradiction in M∂ which converges to E0, and hence, E0 is acyclic in M∂. By Theorem 1.3.1 and Remark 1.3.1 in [35], we obtain that P is uniformly persistent with respect to X0,∂X0. It then follows from Theorem 3.1.1 in [35] that the solution of (3) is uniformly persistent.
Moreover, it follows from Theorem 1.3.6 in [35] that the Poincaree´ map P has a fixed point T^0,L^0,I^0,V^I0∈X0. Then, we see that T^0>0. If not, suppose T^0=0, from the first equation of model (3), where T^t satisfies(28)dT^dt≥λ−dTT^−β1tVI+β2tIT^.
It follows from the comparison theorem that(29)T^t≥e−∫0tasdsT^0+λ∫0te−∫0saτdτds=λe−∫0tasds∫0te−∫0saτdτds,∀t>0,where at=dT+β1tVIt+β2tIt. Then, we have(30)T^nω≥λe−∫0nωasds∫0ωe−∫0saτdτds>0,n=1,2,3,….
The periodicity ofT^t implies T^0=T^nω=0, which is a contradiction. Thus, T^0>0. Hence, T^0,L^0,I^0,V^I0 is a positive ω-periodic solution of model (3).


### 3.2. Analysis of the Autonomous Model

If there no drug therapies, i.e.,ηrtt=ηp1t=ηp2t=0 or drug therapies are constants, then model (3) becomes an autonomous model. Without loss of generality, we assume drug therapy is constant. Then, model (3) leads to the following autonomous model:(31)T′t=λ−dTTt−β1TVI1+m1VI−β2TI1+m2I,L′t=pβ1TVI1+m1VI+pβ2TI1+m2I−α+dLL,I′t=1−pβ1TVI1+m1VI+1−pβ2TI1+m2I+αL−dII,VI′t=NdI1−ηp2I−dVVIt,where β1=k11−ηrt and β2=k21−ηp1. Clearly, model (31) has an infection-free equilibrium E0=T0,0,0,0 with T0=λ/dT. Furthermore, we can obtain the basic reproduction number of model (31):(32)ℛ0=1−p+αpα+dLN1−ηp2β1T0dV+1−p+αpα+dLβ2T0dI=ℛ01+ℛ02.Next, we will show that there exists a unique infection equilibriumE∗=T∗,L∗,I∗,VI∗. For convenience, let fT,VI=TVI/1+m1VI and gT,I=TI/1+m2I. It is easy to see that T∗,L∗,I∗,VI∗ satisfies the following equations:(33)λ−dTT∗=β1T∗fT∗,VI∗+β2gT∗,I∗,pβ1T∗fT∗,VI∗+pβ2gT∗,I∗=α+dLL∗,1−pβ1T∗fT∗,VI∗+1−pβ2gT∗,I∗+αL∗=dII∗,N1−ηp2dII∗=dVVI∗,Then, we can obtain(34)T∗=1dTλ−α+dLpL∗,I∗=α+dL−pdLdIpL∗,VI∗=N1−ηp2α+dL−pdLpdVL∗.AndL∗ satisfies the following equation:(35)β1f1dTλ−α+dLpL∗,N1−ηp2α+dL−pdLpdVL∗+β2g1dTλ−α+dLpL∗,N1−ηp2α+dL−pdLpdVL∗=α+dLpL∗.SinceT∗>0, which implies 0<L∗<λp/α+dL, for L∈0,λp/α+dL, let(36)HL=β1f1dTλ−α+dLpL,N1−ηp2α+dL−pdLpdVL+β2g1dTλ−α+dLpL,N1−ηp2α+dL−pdLpdVL−α+dLpL,Then, we have(37)H0=0,Hλpα+dL=−λ<0,H′0=α+dLpℛ0−1>0forℛ0>1.Thus, there exists aL∗∈0,λp/α+dL. Consequently, model (31) admits an infection equilibrium E∗=T∗,L∗,I∗,VI∗. Furthermore, by calculation, we have(38)H′L∗=−α+dLpdTβ1∂fT∗,VI∗∂T+N1−ηp2α+dL−pdLpdVβ1∂fT∗,VI∗∂VI−α+dLpdTβ2∂gT∗,I∗∂T+α+dL−pdLpdIβ2∂gT∗,I∗∂I−α+dLp=−α+dLpdTβ1∂fT∗,VI∗∂T+VI∗L∗β1∂fT∗,VI∗∂VI−α+dLpdTβ2∂gT∗,I∗∂T+I∗L∗β2∂gT∗,I∗∂I−1L∗β1fT∗,VI∗+β2gT∗,VI∗=−α+dLpdTβ1VI∗1+m1VI∗+β2I∗1+m2I∗−m1β1T∗VI∗2L∗1+m1VI∗2−m2β2T∗I∗2L∗1+m2I∗2<0,which implies that HL is decreasing at each of its root. If there exists more than one root of HL=0, then it must exists a root L^ such that H′L^≥0, which leads to a contradiction. Hence, we claim that there exists a unique infection equilibrium E∗=T∗,L∗,I∗,VI∗ for ℛ0>1.By constructing suitable Lyapunov functions, we can show that the corresponding infection-free equilibriumE0 and infection equilibrium E∗ of model (31) are globally asymptotically stable.Theorem 5.
Ifℛ0≤1, then the infection-free equilibrium E0 of model (31) is globally asymptotically stable.Proof.
Define(39)Ut=ΥT−T0−T0lnTT0+αα+dLL+I+1−ℛ02N1−ηp2VI,where Υ=αp/α+dL+1−p.
By a tedious computation, we have(40)U′t=ΥdT01−TT01−T0T+Υβ1T0VI1+m1VI+Υβ2T0I1+m2I−ℛ02dII−1−ℛ02N1−ηp2dVVI≤ΥdT01−TT01−T0T+Υβ1T0VI+Υβ2T0I−ℛ02dII−1−ℛ02N1−ηp2dVVI=ΥdT01−TT01−T0T+dVVIN1−ηp2ℛ0−1.
Clearly, ifℛ0≤1, then U′t≤0. Moreover, by LaSalle’s invariance principle, one can easy to show that the infection-free equilibrium E0 is globally asymptotically stable.Theorem 6.
Ifℛ0>1, then the infection equilibrium E∗ of model (31) is globally asymptotically stable.Proof.
Define(41)Gt=ΥT−T∗−T∗lnTT∗+αα+dLL−L∗−L∗lnLL∗+I−I∗−I∗lnII∗+Υβ1fT∗,VI∗N1−ηp2dII∗VI−VI∗−VI∗lnVIVI∗,where Υ=αp/α+dL+1−p. Then, combining (33) and by a tedious computation yields(42)G′t=dTT∗Υ1−T∗T1−TT∗+αpα+dLβ1fT∗,VI∗×4−T∗T−LI∗L∗I−IVI∗I∗VI−fT,VIL∗fT∗,VI∗L−VIVI∗+T∗fT,VITfT∗VI∗+1−pβ1fT∗,VI∗3−T∗T−VI∗IVII∗−I∗fT,VIIfT∗,VI∗+T∗fT,VITfT∗,VI∗−VIVI∗+αpα+dLβ2gT∗,I∗3−T∗T−LI∗L∗I−L∗gT,ILgT∗,I∗+T∗gT,ITgT∗,I∗−II∗+1−pβ2gT∗,I∗2−T∗T−gT,II∗gT∗,I∗I+T∗gT,ITgT∗,I∗−II∗,=dTT∗Υ1−T∗T1−TT∗+Υβ1fT∗,VI∗φT∗T+φTfT∗,VI∗VIT∗fT,VIVI∗+φVI∗IVII∗−m1VI−VI∗2VI∗1+m1VI∗1+m1VI+αpα+dLβ1fT∗,VI∗φI∗LIL∗+φfT,VIL∗fT∗,VI∗L+1−pβ1fT∗,VI∗×φI∗fT,VIIfT∗,VI∗+Υβ2gT∗,I∗φT∗T+φTgT∗,I∗IT∗gT,II∗−m2I−I∗2I∗1+m2I1+m2I∗+αpα+dLβ2gT∗,I∗φgT,IL∗gT∗,I∗L+φLI∗L∗I+1−pβ2gT∗,I∗φgT,II∗gT∗,I∗I≤0,where φx=1+lnx−x with maximum value φ1=0 for x>0. Then, it follows from the LaSalle’s invariance principle that one can show E∗ is globally asymptotically stable. This completes the proof.Remark 1.
It follows from the above analysis of the model that the saturated incidence ratesVI/1+m1VI and I/1+m2I can be extended to a more general form fV and gI with some conditions as in [27]. Thus, the saturated incidence rates of model (3) can be regarded as a special case.


## 3.1. Stability and Persistence of the Disease

In this section, we will investigate the global asymptotic stability of infection-free equilibrium and the disease persistence by regardingℛ0 as a threshold parameter.Theorem 3.
Ifℛ0<1, then the infection-free equilibrium E0 is globally asymptotically stable, and it is unstable for ℛ0>1.Proof.
It follows from Theorem2 that if ℛ0<1, then E0 is locally asymptotically stable and E0 is unstable when ℛ0>1. Hence, it is sufficient to show that E0 is global attractive for ℛ0<1.
From the first equation of (3) and nonnegativity of the solutions, we then have dT/dt≤λ−dTT, which implies that ∀ε>0; there exists t˜>0 such that Tt≤T0+ε, t>t˜.
Consider the following auxiliary system:(14)L˜′t=pβ2tT0+εI˜+pβ1tT0+εV˜I−α+dLL˜,I˜′t=1−pβ2tT0+εI˜+1−pβ1tT0+εV˜I+αL˜−dII˜,V˜I′t=NdIatI˜−dVV˜It,which is equivalent to(15)L˜′I˜′V˜I′=F−VL˜I˜V˜I+εMtL˜I˜V˜I,where(16)Mt=0pβ2tpβ1t01−pβ2t1−pβ1t000.
It then follows from Lemma1 that there exists a positive ω-periodic function vt=v1t,v2t,v3t such that eμtvt is a solution of (14), where μ=1/ωlnrΦF−V+εMω. Choose t¯>t˜ and a real number α1>0 such that L˜t¯,I˜t¯,V˜It¯T≤α1v0, which implies that(17)L˜t¯,I˜t¯,V˜It¯T≤α1eμt−t¯vt−t¯,t≥t¯.
The comparison principle yields that(18)Lt,It,VItT≤α1eμt−t¯vt−t¯,t≥t¯.
Recall Theorem2 that ℛ0<1 if and only if rΦF−Vω<1. Since the continuity of the spectrum for matrices [34], then choose ε>0 small enough such that rΦF−V+εMω<1, which implies that μ<0. Then, we have Lt,It,VItT⟶0 as t⟶∞. Furthermore, it follows from the first equation of model (3) and the theory of asymptotically periodic semiflows [35] that limt⟶∞Tt=T0. Thus, E0 is globally attractive.Theorem 4.
Ifℛ0>1, then there exists an ε>0 such that any solution Tt,Lt,It,VIt of model (3) with initial values T0,L0,I0,VI0=T0,L0,I0,VI0∈ℝ+×Intℝ+3; the solution of (3) satisfies liminft⟶∞Tt,Lt,It,VIt≥ε,ε,ε,ε and admits at least one positive periodic solution.Proof.
Let(19)X=ℝ+4,X0=ℝ+×Intℝ+3,∂X0=XX0.
Define Poincaree´ map P:ℝ+4⟶ℝ+4, satisfying Px0=uω,x0, ∀x0∈ℝ+4, with ut,x0 as the unique solution of (3) satisfying u0,x0=x0.
We first show thatP is uniformly persistent with respect to X0,∂X0. It is easy to see that X and X0 are positively invariant. Moreover, ∂X is a relatively closed set in X. Recall Theorem 1 that the solutions of model (3) are uniformly and ultimately bounded; thus, the semiflow P is point dissipative on ℝ+4, and P:ℝ+4⟶ℝ+4 is compact. Consequently, it follows from [36] that the semiflow P admits a global attractor, which attracts every bounded set in ℝ+4.
Define(20)M∂=T0,L0,I0,VI0∈∂X0:PmT0,L0,I0,VI0∈∂X0,∀m≥0.
Then, we claim thatM∂=T,0,0,0:T≥0. In fact, it is obvious that T,0,0,0:T≥0⊆M∂. For any T0,L0,I0,VI0∈∂X0/T,0,0,0:T≥0, considering the following cases: (1) L0=0,I0>0,VI0>0, (2) L0>0,I0=0,VI0>0, (3) L0>0,I0>0,VI0=0, (4) L0=I0=0,V0>0, (5) L0=VI0=0,I0>0, and (6) L0>0,I0=VI0=0. For case (1), we have dL/dt|t=0=pβ10T0VI0+pβ20T0I0>0, which implies that Tt,Lt,It,VIt∉∂X0, for 0<t≪1; then, Tt,Lt,It,VIt∉M∂. Similarly, for the other cases, it also has the same result; here, we omit the proof. Thus, for any T0,L0,I0,VI0∉T,0,0,0:T≥0, T0,L0,I0,VI0∉M∂, it indicates that M∂⊆T,0,0,0:T≥0.
Clearly,E0 is one fixed point of P in M∂. If Tt,Lt,It,VIt is a solution of model (3) from M∂, it then follows from that model (3) that Tt⟶T0,Lt⟶0,It⟶0,VIt⟶0 as t⟶∞.
Next, we will show that if the invariant setE0 is isolated, then E0 is an acyclic covering. To do this, it needs to prove any solution of model (3) initiating from M∂ will remain into M∂, which can be obtained easily. The isolated invariance of E0 will follow proof.
Now, we need to prove thatWsE0∩X0=∅. Denote x0=T0,L0,I0,VI0∈X0. Since the continuity of solutions with respect to the initial values, thus for ∀ε>0, there exists δ0>0 such that, for all x0∈X0 with x0−E0≤δ0, yields(21)ut,x0−ut,E0≤ε,∀t∈0,ω.
Then, we claim that(22)limsupm⟶∞dPmx0,E0≥δ0.
If it is not true, then we have(23)limsupm⟶∞dPmx0,E0<δ0.
For somex0∈X0, without loss of generality, we suppose that limsup/limits_t⟶∞dPmx0,E0<δ0, ∀m>0. Then, we can obtain that(24)ut,Pmx0−ut,E0∥≤ε,∀t∈0,ω.
For anyt≥0, let t=mω+t1, where t1∈0,ω and m=t/ω, which is the greatest integer less than or equal to t/ω. Then, we have(25)ut,Pmx0−ut,E0=ut1,Pmx0−ut1,E0≤ε,∀t∈0,ω.
SetTt,Lt,It,VIt=ut,x0; then, we have T0−ε≤T≤T0+ε, 0≤L≤ε, 0≤I≤ε, and 0≤VI≤ε, for t≥0. Then, T/1+m1VI≥T0−ε/1+m1ε=T0−ε1+m1T0/1+m1ε≥T0−ε1+m1T0 and T/1+m2I≥T0−ε/1+m2ε=T0−ε1+m2T0/1+m2ε≥T0−ε1+m2T0. It follows from model (3) that(26)L′t≥pβ1tT0−ε1+m1T0VI+pβ2tT0−ε1+m2T0I−α+dLL,I′t≥1−pβ1tT0−ε1+m1T0VI+β2tT0−ε1+m2T0I+αL−dII,VI′t=NdIatI−dVVIt.
Set(27)Mε=0ε1+m2T0pβ2tε1+m1T0pβ1t0ε1+m2T01−pβ2tε1+m1T01−pβ1t000.
It follows from Theorem2 that rΦF−Vω>1; then, we can select ε>0 small enough such that rΦF−V−Mεω>1. It follows from Lemma 1 and the standard comparison principle that there exists a positive ω-periodic function v¯t=v¯1t,v¯2t,v¯3tT such that Qt≥v¯teμ1t, where Qt=Lt,It,VItT and μ1=1/ωlnrΦF−V−Mεω>0, which implies that lim/limits_t⟶∞Lt,It,VIt=∞; this is a contradiction in M∂ which converges to E0, and hence, E0 is acyclic in M∂. By Theorem 1.3.1 and Remark 1.3.1 in [35], we obtain that P is uniformly persistent with respect to X0,∂X0. It then follows from Theorem 3.1.1 in [35] that the solution of (3) is uniformly persistent.
Moreover, it follows from Theorem 1.3.6 in [35] that the Poincaree´ map P has a fixed point T^0,L^0,I^0,V^I0∈X0. Then, we see that T^0>0. If not, suppose T^0=0, from the first equation of model (3), where T^t satisfies(28)dT^dt≥λ−dTT^−β1tVI+β2tIT^.
It follows from the comparison theorem that(29)T^t≥e−∫0tasdsT^0+λ∫0te−∫0saτdτds=λe−∫0tasds∫0te−∫0saτdτds,∀t>0,where at=dT+β1tVIt+β2tIt. Then, we have(30)T^nω≥λe−∫0nωasds∫0ωe−∫0saτdτds>0,n=1,2,3,….
The periodicity ofT^t implies T^0=T^nω=0, which is a contradiction. Thus, T^0>0. Hence, T^0,L^0,I^0,V^I0 is a positive ω-periodic solution of model (3).


## 3.2. Analysis of the Autonomous Model

If there no drug therapies, i.e.,ηrtt=ηp1t=ηp2t=0 or drug therapies are constants, then model (3) becomes an autonomous model. Without loss of generality, we assume drug therapy is constant. Then, model (3) leads to the following autonomous model:(31)T′t=λ−dTTt−β1TVI1+m1VI−β2TI1+m2I,L′t=pβ1TVI1+m1VI+pβ2TI1+m2I−α+dLL,I′t=1−pβ1TVI1+m1VI+1−pβ2TI1+m2I+αL−dII,VI′t=NdI1−ηp2I−dVVIt,where β1=k11−ηrt and β2=k21−ηp1. Clearly, model (31) has an infection-free equilibrium E0=T0,0,0,0 with T0=λ/dT. Furthermore, we can obtain the basic reproduction number of model (31):(32)ℛ0=1−p+αpα+dLN1−ηp2β1T0dV+1−p+αpα+dLβ2T0dI=ℛ01+ℛ02.Next, we will show that there exists a unique infection equilibriumE∗=T∗,L∗,I∗,VI∗. For convenience, let fT,VI=TVI/1+m1VI and gT,I=TI/1+m2I. It is easy to see that T∗,L∗,I∗,VI∗ satisfies the following equations:(33)λ−dTT∗=β1T∗fT∗,VI∗+β2gT∗,I∗,pβ1T∗fT∗,VI∗+pβ2gT∗,I∗=α+dLL∗,1−pβ1T∗fT∗,VI∗+1−pβ2gT∗,I∗+αL∗=dII∗,N1−ηp2dII∗=dVVI∗,Then, we can obtain(34)T∗=1dTλ−α+dLpL∗,I∗=α+dL−pdLdIpL∗,VI∗=N1−ηp2α+dL−pdLpdVL∗.AndL∗ satisfies the following equation:(35)β1f1dTλ−α+dLpL∗,N1−ηp2α+dL−pdLpdVL∗+β2g1dTλ−α+dLpL∗,N1−ηp2α+dL−pdLpdVL∗=α+dLpL∗.SinceT∗>0, which implies 0<L∗<λp/α+dL, for L∈0,λp/α+dL, let(36)HL=β1f1dTλ−α+dLpL,N1−ηp2α+dL−pdLpdVL+β2g1dTλ−α+dLpL,N1−ηp2α+dL−pdLpdVL−α+dLpL,Then, we have(37)H0=0,Hλpα+dL=−λ<0,H′0=α+dLpℛ0−1>0forℛ0>1.Thus, there exists aL∗∈0,λp/α+dL. Consequently, model (31) admits an infection equilibrium E∗=T∗,L∗,I∗,VI∗. Furthermore, by calculation, we have(38)H′L∗=−α+dLpdTβ1∂fT∗,VI∗∂T+N1−ηp2α+dL−pdLpdVβ1∂fT∗,VI∗∂VI−α+dLpdTβ2∂gT∗,I∗∂T+α+dL−pdLpdIβ2∂gT∗,I∗∂I−α+dLp=−α+dLpdTβ1∂fT∗,VI∗∂T+VI∗L∗β1∂fT∗,VI∗∂VI−α+dLpdTβ2∂gT∗,I∗∂T+I∗L∗β2∂gT∗,I∗∂I−1L∗β1fT∗,VI∗+β2gT∗,VI∗=−α+dLpdTβ1VI∗1+m1VI∗+β2I∗1+m2I∗−m1β1T∗VI∗2L∗1+m1VI∗2−m2β2T∗I∗2L∗1+m2I∗2<0,which implies that HL is decreasing at each of its root. If there exists more than one root of HL=0, then it must exists a root L^ such that H′L^≥0, which leads to a contradiction. Hence, we claim that there exists a unique infection equilibrium E∗=T∗,L∗,I∗,VI∗ for ℛ0>1.By constructing suitable Lyapunov functions, we can show that the corresponding infection-free equilibriumE0 and infection equilibrium E∗ of model (31) are globally asymptotically stable.Theorem 5.
Ifℛ0≤1, then the infection-free equilibrium E0 of model (31) is globally asymptotically stable.Proof.
Define(39)Ut=ΥT−T0−T0lnTT0+αα+dLL+I+1−ℛ02N1−ηp2VI,where Υ=αp/α+dL+1−p.
By a tedious computation, we have(40)U′t=ΥdT01−TT01−T0T+Υβ1T0VI1+m1VI+Υβ2T0I1+m2I−ℛ02dII−1−ℛ02N1−ηp2dVVI≤ΥdT01−TT01−T0T+Υβ1T0VI+Υβ2T0I−ℛ02dII−1−ℛ02N1−ηp2dVVI=ΥdT01−TT01−T0T+dVVIN1−ηp2ℛ0−1.
Clearly, ifℛ0≤1, then U′t≤0. Moreover, by LaSalle’s invariance principle, one can easy to show that the infection-free equilibrium E0 is globally asymptotically stable.Theorem 6.
Ifℛ0>1, then the infection equilibrium E∗ of model (31) is globally asymptotically stable.Proof.
Define(41)Gt=ΥT−T∗−T∗lnTT∗+αα+dLL−L∗−L∗lnLL∗+I−I∗−I∗lnII∗+Υβ1fT∗,VI∗N1−ηp2dII∗VI−VI∗−VI∗lnVIVI∗,where Υ=αp/α+dL+1−p. Then, combining (33) and by a tedious computation yields(42)G′t=dTT∗Υ1−T∗T1−TT∗+αpα+dLβ1fT∗,VI∗×4−T∗T−LI∗L∗I−IVI∗I∗VI−fT,VIL∗fT∗,VI∗L−VIVI∗+T∗fT,VITfT∗VI∗+1−pβ1fT∗,VI∗3−T∗T−VI∗IVII∗−I∗fT,VIIfT∗,VI∗+T∗fT,VITfT∗,VI∗−VIVI∗+αpα+dLβ2gT∗,I∗3−T∗T−LI∗L∗I−L∗gT,ILgT∗,I∗+T∗gT,ITgT∗,I∗−II∗+1−pβ2gT∗,I∗2−T∗T−gT,II∗gT∗,I∗I+T∗gT,ITgT∗,I∗−II∗,=dTT∗Υ1−T∗T1−TT∗+Υβ1fT∗,VI∗φT∗T+φTfT∗,VI∗VIT∗fT,VIVI∗+φVI∗IVII∗−m1VI−VI∗2VI∗1+m1VI∗1+m1VI+αpα+dLβ1fT∗,VI∗φI∗LIL∗+φfT,VIL∗fT∗,VI∗L+1−pβ1fT∗,VI∗×φI∗fT,VIIfT∗,VI∗+Υβ2gT∗,I∗φT∗T+φTgT∗,I∗IT∗gT,II∗−m2I−I∗2I∗1+m2I1+m2I∗+αpα+dLβ2gT∗,I∗φgT,IL∗gT∗,I∗L+φLI∗L∗I+1−pβ2gT∗,I∗φgT,II∗gT∗,I∗I≤0,where φx=1+lnx−x with maximum value φ1=0 for x>0. Then, it follows from the LaSalle’s invariance principle that one can show E∗ is globally asymptotically stable. This completes the proof.Remark 1.
It follows from the above analysis of the model that the saturated incidence ratesVI/1+m1VI and I/1+m2I can be extended to a more general form fV and gI with some conditions as in [27]. Thus, the saturated incidence rates of model (3) can be regarded as a special case.


## 4. Numerical Simulations

In this section, some numerical simulations are carried out to explain the obtained theoretical results. Most of these parameter values are taken from [6, 13].Case 1.
Letλ=100, dT=0.02, k1=2.4×10−6, k2=1.5×10−4, α=0.1, N=1500, p=0.01, dV=23, dI=0.4, dL=4×10−3, m1=0.01, m2=0.01, ηrt=0.6−0.3cosπt/12, ηp1=0.7−0.2cosπt/12, and ηp2=0.6−0.25cosπt/12; then, we can obtain ℛ0=0.7875<1. The simulation shows that the infection-free equilibrium E0=5000,0,0,0 is globally asymptotically stable, which implies disease dies out. Figure 1 validates the above analysis.Figure 1
The results show that the infection-free equilibriumE0 is globally asymptotically stable when ℛ0<1.Case 2.
Letλ=100, dT=0.02, k1=2.4×10−4, k2=1×10−4, α=0.1, N=2000, p=0.01, dV=23, dI=0.4, dL=4×10−3, m1=0.01, m2=0.01, ηrt=0.6−0.3cosπt/12, ηp1=0.7−0.2cosπt/12, and ηp2=0.6−0.25cosπt/12; then, we have ℛ0=17.0641>1. The theoretical results show that the model admits a positive periodic solution and disease keeps persistent in the host. Figure 2 confirms this conclusion.
Figure3 shows the impact of cell-to-cell transmission k2 on the dynamics of the model, and the other parameters are the same with Figure 2. The results imply that the peak level of the density of latent cells, infected cells, and viral load increase as k2 increases. Hence, cell-to-cell transmission existing may contribute to viral persistence and underevaluate the risk of disease spreading for without considering cell-to-cell transmission. Thus, cell-to-cell transmission will be a barrier to curing the viral infection.
Figure4 shows that the effects of PIs on the dynamics of the model by blocking cell-to-cell transmission, and the other parameters are the same with Figure 2. By varying the drug efficacy ηp1 of PIs for blocking cell-to-cell transmission from the baseline value ηp1=0.7−0.2cosπt/12 to 50% and 0%, the numerical results imply that the peak level of the density of latent cell, infected cells, and viral load decrease under PIs’ blocking cell-to-cell transmission. This implies that increase of the efficacy of PIs for blocking cell-to-cell transmission may contribute to weakening the severity of the viral infection.Figure 2
The results shows that model (3) exists as a periodic solution when ℛ0>1.Figure 3
The effect of cell-to-cell transmissionk2 on the dynamics of model (3). This shows that the existence of cell-to-cell transmission is benefit for viral infection.Figure 4
The effect of PIs for blocking cell-to-cell transmissionηp1 on the dynamics of model (3). The results show that PIs’ blocks the cell-to-cell transmission (i.e., ηp1≠0) may contribute to decreasing the peak level of the density of latent cells, infected cells, and viral load on some recent.


## 5. Summary and Discussion

In this study, a time-periodic viral infection model with cell-cell transmission was investigated. We have shown that the infection-free equilibriumE0 is globally asymptotically stable if ℛ0<1 which implies that infection will be eradicated, and the infection will persistent when ℛ0>1. Furthermore, for the corresponding autonomous model, we have shown that the corresponding equilibria are globally asymptotically stable by applying the method of Lyapunov function. The results imply that cell-to-cell transmission existing may contribute to viral persistence and be a barrier to curing the viral infection. At the same time, increasing the efficacy of PIs for blocking cell-to-cell transmission is beneficial to weakening the severity of the viral infection.Note that only a constant recruitment of uninfected T-cells has been considered in our model. However, T-cells can also be created by proliferation of existing T-cells with a logistic form instead of a constant recruitment [19] and the incidence rates can be a more general form fV and gI [27], which formulated a further work. Furthermore, immune response and dynamics of the drug are also a good choice to extend the current work, which will be another future work.



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*Source: 2289138-2022-03-21.xml*