# Dynamic Consistent NSFD Scheme for a Delayed Viral Infection Model with Immune Response and Nonlinear Incidence

Authors: Jinhu Xu; Yan Geng
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

In this paper, a discrete-time model has been proposed by applying nonstandard finite difference (NSFD) scheme to solve a delayed viral infection model with immune response and general nonlinear incidence. It is shown that the discrete model has equilibria which are exactly the same as those of the original continuous model. Using discrete-time analogue of Lyapunov functionals, the global asymptotic stability of the equilibria of the discrete model is fully determined by the basic reproduction number of the virus and immune response,R0 and R1, with no restriction on the time step size, which implies that the NSFD scheme preserves the qualitative dynamics of the corresponding continuous model.


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## Body

## 1. Introduction

Since samples cannot always be taken frequently from patients, or detection techniques of the virus may not be accurate, testing specific hypotheses based on clinical data is a challengeable task, which justifies the critical role played by mathematical models in describing the dynamics inside the host of various infectious diseases such as HBV, HCV, and HIV. Over the past several decades, many models for studying infectious disease have been proposed and studied. The classical model for within host virus dynamics is a system which includes three ordinary differential equations [1, 2]. However, to take some features into consideration of a real system, like time delay, age structure, and so on, many literatures have been proposed and studied (see [3–8] and references therein). For example, Manna [8] considered a delayed HBV infection model with HBV DNA-containing capsids which takes the following form: (1)H′=s-μHt-kHtVt,I′=kHt-τ1Vt-τ1-δIt-pItZt,D′=aIt-τ2-β+δDt,V′=βDt-cVt,Z′=qItZt-σZt,where H(t),I(t),D(t), V(t), and Z(t) denote the densities of the uninfected hepatocytes, infected hepatocytes, intracellular HBV DNA-containing capsids, the virions, and CTL cells at time t, respectively. The hepatocytes are assumed to be produced from a source at rate s, have a natural death rate μ, and get infected by the virions at a rate k. δ is the death rate of infected hepatocytes and capsids. a represents the rate of production of HBV DNA-containing capsids. Capsids lead to viral replication at the rate β, and c is the nature death rate of the virions. p is the rate the infected hepatocytes are removed by CTLs while q accounts for the CTL responsiveness and σ represents decay rate for CTLs in absence of stimulation. τ1 denotes the time needed in the production of productively infected hepatocytes from the uninfected ones; τ2 means the time spent in the production of matured intracellular HBV DNA-containing capsids which in turn contributes to the production of virions. The global asymptotic stability of the equilibria of model (1) has been investigated in [8] by constructing Lyapunov functionals.Note that the bilinear incidence rate is a simple description of the infection in model (1). However, as mentioned in [9], a general incidence rate may help us to gain the unification theory by the omission of unessential details. For more details about nonlinear incidence rates, we refer to see [10–12] and references cited in. Hence, inspired by the aforementioned literatures, we consider the following delayed model with general nonlinear incidence: (2)H′=s-μHt-kHtfVt,I′=kHt-τ1fVt-τ1-δIt-pItZt,D′=aIt-τ2-β+δDt,V′=βDt-cVt,Z′=qItZt-σZt.Here, the incidence is assumed to be the nonlinear responses to the concentration of virions taking the form kHf(V), where f(V) denote the force of infection by virus particles and satisfy the following properties [13]: (3)f0=0,f′V>0,f′′V≤0.Based on condition (3), it follows from the Mean Value Theorem that (4)f′VV≤fV≤f′0V.Epidemiologically, condition (3) indicates that (i) the disease can not spread if there is no infection; (ii) the incidences kHf(V) become faster as the densities of the virions increase; (iii) the per capita infection rates by virions will slow down due to certain inhibition effect since (4) implies that fV/V′≤0.Obviously, the incidence rate with condition (3) contains the bilinear and the saturation incidences.The initial conditions for model (2) are (5)Hθ=ψ1θ,Iθ=ψ2θ,Dθ=ψ3θ,Vθ=ψ4θ,Zθ=ψ5θ,ψiθ≥0,θ∈-τ,0,ψi0>0i=1,2,3,4,5,where τ=max{τ1,τ2} and (ψ1(θ),ψ2(θ),ψ3(θ),ψ4(θ),ψ5(θ))∈C([-τ,0],R+5), the Banach space of continuous functions mapping the interval [-τ,0] into R+5 with R+5={(x1,x2,x3,x4,x5):xi≥0,i=1,2,3,4,5}.In order to investigate the dynamics of the solutions for a model like (1), we need to get the exact solution for the model, which is one of the most important tasks of mathematical modelling. However, this is very difficult or even impossible to be determined. Hence, researchers seek numerical ones instead. However, how to select a proper discrete method so that the global properties of solutions of the corresponding continuous models can be efficiently preserved is still an open problem [14]. Recently, Mickens has made an attempt in this regard, by proposing a robust nonstandard finite difference (NSFD) scheme [15, 16], which has been widely employed in the study of different kinds of epidemic models and one important advantage of Mickens’ method is that it can be more efficient in preserving the global dynamics to the corresponding continuous epidemic models [10, 17–23]. However, there is no result about discrete viral infection model with time delays and immune response. Therefore, motivated by [15, 16], we obtain from model (2) (6)Hn+1-HnΔt=s-μHn+1-kHn+1fVn,In+1-InΔt=kHn-m1+1fVn-m1-δIn+1-pIn+1Zn+1,Dn+1-DnΔt=aIn-m2+1-β+δDn+1,Vn+1-VnΔt=βDn+1-cVn+1,Zn+1-ZnΔt=qIn+1Zn+1-σZn+1,where Δt>0 is the time step size and (Hn,In,Dn,Vn,Zn) are the approximations of the solution (H(tn),I(tn),D(tn),V(tn),Zn) of model (2) at the discrete-time point tn=Δtn (n∈N) and m1,m2∈N are constant integers satisfying τ1=m1Δt, τ2=m2Δt.The discrete initial condition of model (6) is given as (7)Hs=ϕs1≥0,Is=ϕs2≥0,Ds=ϕs3≥0,Vs=ϕs4≥0,Zs=ϕs5≥0,∀s=-m,-m+1,…,0,where m=max{m1,m2} and ϕ0(i)>0(i=1,2,3,4,5).In this paper, we will present an affirmative answer that the discrete model (6) which derived by utilizing Mickens’ method can efficiently preserve the global properties to the original continuous model (2). The rest of this paper is organized as follows. We first study the global dynamics of the continuous system (2) in Section 2. In Section 3, we investigate the global dynamics of discrete system (6). A brief conclusion ends the paper.


## 2. Global Dynamics of Model (2)

### 2.1. Preliminaries

The following result established the positivity and boundedness of solutions of model (2).Theorem 1.
Let(H(t),I(t),D(t),V(t),Z(t)) be the solution of model (2) satisfying conditions (5). Then H(t),I(t),D(t),V(t) and Z(t) are all nonnegative and bounded for all t>0.Proof.
First, we prove thatH(t)>0 for all t≥0. Assume the contrary and let T0>0 such that H(T0)=0. Then from the first equation of system (2), we have H′(t)=λ>0. Therefore, H(t)<0 for t∈(T0-ε,T0) and ε>0 is sufficiently small. This contradicts with the fact of H(t)>0 for t∈[0,T0). It follows that H(t)>0 for t≥0. Moreover, it follows from (2) that (8)It=I0e-∫0tδ+pZθdθ+∫0te-∫θtδ+pZθdθkHθ-τ1fVθ-τ1dθ,Dt=D0e-β+δt+∫0taIθ-τ2eβ+δθ-tdθ,Vt=V0e-ct+∫0tβDθecθ-tdθ,Zt=Z0e-∫0tσ-qIθdθ.Then the nonnegative immediately follows from the above integral forms and (5).
Next, we show the boundedness of the solution. Define(9)Gt=Ht+It+τ1+δ2aDt+τ1+τ2+δ2aVt+τ1+τ2+pqZt+τ1,and d0=minμ,δ/2,c,σ. It then follows that (10)G′t=s-μHt-δ2It+τ1-δ22Dt+τ1+τ2-δc2aVt+τ1+τ2-σpqZt+τ1≤s-d0Gt.This implies that G(t) is bounded and so are H(t),I(t),D(t),V(t), and Z(t). This completes the proof.


### 2.2. Steady States

Obviously, the model (2) always has an infection-free equilibrium E0=(H0,0,0,0,0) with H0=s/μ. This is the only biologically meaningful equilibrium if (11)R0=aksβf′0δcμβ+δ<1.At an equilibrium of model (2) we have (12)s=μH+kHfV,kHfV=δI+pIZ,aI=β+δD,βD=cV,qI-σZ=0.It is clear that model (2) only has the following two possible equilibria except the E0, that is, E1=(H1,I1,D1,V1,0) and E2=(H2,I2,D2,V2,Z2), where H1,I1,D1,V1,H2,I2,D2,V2,Z2 are all strictly positive.IfZ=0, then the existence of E1=(H1,I1,D1,V1,0) is equivalent to the existence of positive solution (H1,I1,D1,V1) of the following equations: (13)s=μH+kHfV,kHfV=δI,aI=β+δD,βD=cV.From the last three equations of (13) we have (14)s-μH=δcβ+δVaβ,I=β+δcVaβ,D=cVβ.This means that, in order to have H≥0 and V>0 at an equilibrium, we must have V∈(0,aβs/δc(β+δ)]. Further, substituting I into the second equation of (13) gives (15)H=δcβ+δVakβfV.Then substituting H into the first equation of (13), direct calculation yields (16)0=s-μδcβ+δVaβkfV-δcβ+δVaβ≕FV.For F(V) with V>0, it then follows from (4) that (17)F′V=-δcβ+δaβ-μδcβ+δakβfV-f′VV<0.Moreover, we obtain that (18)limV→0+FV=μδcβ+δaβkf′0R0-1,Faβsδcβ+δ=-μskfaβs/δcβ+δ<0.Therefore, there exists an infection equilibrium without immunity E1=(H1,I1,D1,V1,0) when R0>1.Define(19)R1=qksfβaσ/qcβ+δδσμ+kfβaσ/qcβ+δwhich represents the immune response activation number and determines whether or not a persistent immune response can be established. If Z≠0, then from (12) we have (20)H2=sμ+kfV2,I2=σq,D2=aσqβ+δ,V2=aβσqcβ+δ,Z2=δpR1-1.Therefore, the infection equilibrium with immunity E2=(H2,I2,D2,V2,Z2) exists provided R1>0. It follows from the properties of function f(V) that R1<R0. Concluding the above analysis we have the following result.Theorem 2.
For model (2).(i)
IfR0<1, then there exists a unique infection-free equilibrium E0;(ii)
IfR1≤1<R0, then there exists a unique infection equilibrium without immunity E1 besides E0;(iii)
IfR1>1, then there exists a unique infection equilibrium with immunity E2 besides E0 and E1.


### 2.3. Global Dynamics of Model (2)

In this part, we investigate the global asymptotic stability of the equilibria of system (2) by constructing Lyapunov functionals. To this end, we first introduce the following function for the following work, φx=x-1-ln⁡x for x∈(0,+∞), which will be used in Lyapunov functionals. It is easy to show that φ(x)≥φ(1)=0.Theorem 3.
IfR0≤1, then the infection-free equilibrium E0 is globally asymptotically stable.Proof.
We construct a Lyapunov functional as follows:(21)L1=H0φHH0+I+δaD+δβ+δaβV+pqZ+∫t-τ1tkHθfVθdθ+∫t-τ2tIθdθ.To proceed, we denote uτ=u(t-τ) (u∈{H,I,V}, τ∈{τ1,τ2}), for the sake of convenience. Calculating dL1/dt along the solutions of system (2) and applying s=μH0, together with condition (4), yield (22)dL1dt=1-H0Hs-μH-kHfV+kHτ1fVτ1-δI-pIZ+δaaIτ2-β+δD+δβ+δaββD-cV+kHfV-Hτ1fVτ1+δI-Iτ2=μH01-H0H1-HH0+kH0fV-δβ+δcVaβ-pσqZ≤μH01-H0H1-HH0+δcβ+δaβR0-1V-pσqZ.Therefore, if R0≤1, then dL1/dt≤0. Furthermore, it can be shown that the largest invariant subset of dL1/dt=0 is the singleton {E0}. Thus, the infection-free equilibrium E0 is globally asymptotically stable. This completes the proof.To establish the global stability of the infection equilibrium without immunityE1 when R1≤1<R0, we first give the following results.Lemma 4.
Under condition (3), for Vi>0,i=1,2, it holds that (23)fV1-fV2V1-V2>0,(24)fV2V2-fV1V1V1-V2>0,(25)fVfVi-VVi1-fVifV≤0,i=1,2.This Lemma can be easily obtained from the properties of functionf(V). Hence, we omit the proof. Based on Lemma 4, we present the following Lemma.Lemma 5.
Suppose the condition (3) is satisfied and R0>1. Then Hi,Ii,Di,Vi(i=1,2) exist satisfying (26)sgnH2-H1=sgnI1-I2=sgnD1-D2=sgnV1-V2=sgnR1-1.Proof.
First, we claim thatsgn(H2-H1)=sgn(I1-I2). On the contrary, we have sgn(H1-H2)=sgn(I1-I2). It follows from the conditions of the equilibria E1 and E2 that sgn(I1-I2)=sgn(D1-D2)=sgn(V1-V2); furthermore, (27)μH2-H1=kH1fV1-H2fV2=kH1-H2fV1+kH2fV1-fV2.Thus, from (23) in Lemma 4 we get sgn(H2-H1)=sign(H1-H2) which leads to a contradiction. Thus, sign(H2-H1)=sgn(I1-I2).
Next, we will provesgn(R1-1)=sgn(I1-I2). To this end, let (28)R~1=kH2fV2δI2;then it is easy to see that sgn(R1-1)=sgn(R~1-1). Thus, we just need to prove sgn(R~1-1)=sgn(I1-I2). Use the equilibria conditions of E1 and E2, which gives (29)R~-1=kH2fV2δI2-kH1fV1δI1=akβδcβ+δH2fV2V2-H1fV1V1=akβδcβ+δH2-H1fV2V2+H1fV2V2-fV1V1.Thus, it follows from (24) that sgn(R~1-1)=sgn(I1-I2). This completes the proof.Theorem 6.
IfR1≤1<R0, then the infection equilibrium without immunity E1 is globally asymptotically stable.Proof.
Constructing a Lyapunov functionalL2 as follows(30)L2=H1φHH1+I1φII1+δaD1φDD1+δβ+δV1aβφVV1+pqZ+kH1fV1∫t-τ1tφHθfVθH1fV1dθ+δI1∫t-τ2tφIθI1dθ.Take the derivative of L2 along solutions of model (2) and recall that the equilibrium conditions of E1 are(31)s=μH1+kH1fV1,kH1fV1=δI1=δβ+δD1a=δcβ+δV1aβ.Then, we obtain that (32)dL2dt=1-H1Hs-μH-kHfV+1-I1IkHτ1fVτ1-δI-pIZ+δa1-D1DaIτ2-β+δD+δβ+δaβ1-V1VβD-cV+pqqIZ-σZ+kH1fV1HfVH1fV1-Hτ1fVτ1H1fV1+ln⁡Hτ1fVτ1HfV+δI1II1-Iτ2I1+ln⁡Iτ2I=μH11-H1H1-HH1+1-H1HkH1fV1-kHfV+1-I1IkHτ1fVτ1-δI1II1+δa1-D1DaI1Iτ2I1-β+δδD1aDD1+δβ+δaβ1-V1VβD1DD1-cV1VV1+pZI1-σq+kH1fV1HfVH1fV1-Hτ1fVτ1H1fV1+ln⁡Hτ1fVτ1HfV+δI1II1-Iτ2I1+ln⁡Iτ2I=μH11-H1H1-HH1+kH1fV14-H1H-Hτ1fVτ1I1H1fV1I-Iτ2D1ID-DV1D1V+fVfV1-VV1+ln⁡Hτ1fVτ1Iτ2HfVI+pZI1-I2=μH11-H1H1-HH1+kH1fV1-φH1H-φHτ1fVτ1I1H1fV1I-φIτ2D1ID-φDV1D1V+fVfV1-VV1+ln⁡fV1VfVV1+pZI1-I2=μH11-H1H1-HH1+kH1fV1-φH1H-φHτ1fVτ1I1H1fV1I-φIτ2D1ID-φDV1D1V-φfV1VfVV1+1-fV1fVfVfV1-VV1+pZI1-I2.From Lemma 5 we can obtain that R1≤1 is equivalent to I1-I2≤0. Combined with (25) in Lemma 4, it follows that dL2/dt≤0 for all H,I,D,V,Z>0. Furthermore, it can be shown that the largest invariant subset of dL2/dt=0 is the singleton {E1}. Hence, the global asymptotic stability of the infection equilibrium without immunity E1 follows from LaSalle’s invariant principle. This completes the proof.Theorem 7.
IfR1>1, then the infection equilibrium with immunity E2 is globally asymptotically stable.Proof.
Constructing a Lyapunov functionalL3 as follows: (33)L3=H2φHH2+I2φII2+δ+pZ2aD2φDD2+δ+pZ2β+δV2aβφVV2+pqφZZ2+kH2fV2∫t-τ1tφHθfVθH2fV2dθ+δ+pZ2I2∫t-τ2tφIθI2dθ.Take the derivative of L3 along solutions of model (2) and recall that the equilibrium conditions of E2 are (34)s=μ+kfV2H2,I2=σq,kH2fV2=δ+pZ2I2=δ+pZ2β+δD2a=δ+pZ2β+δcV2aβ.Then, we obtain that (35)dL3dt=1-H2Hs-μH-kHfV+1-I2IkHτ1fVτ1-δI-pIZ+δ+pZ2a1-D2DaIτ2-β+δD+δ+pZ2β+δaβ1-V2VβD-cV+pq1-Z2ZqIZ-σZ+kH2fV2HfVH2fV2-Hτ1fVτ1H2fV2+ln⁡Hτ1fVτ1HfV+δ+pZ2I2II2-Iτ2I2+ln⁡Iτ2I=μH21-H2H1-HH2+1-H2HkH2fV2-kHfV+1-I2IkHτ1fVτ1-kH2fV2II2+pIZ2-pIZ+δ+pZ2a1-D2DaI2Iτ2I2-β+δδD2aDD2+δ+pZ2β+δaβ1-V2VβD1DD1-cV1VV1+pq1-Z2ZqIZ-σZ+kH2fV2HfVH2fV2-Hτ1fVτ1H2fV2+ln⁡Hτ1fVτ1HfV+δ+pZ2I2II2-Iτ2I2+ln⁡Iτ2I=μH21-H2H1-HH2+kH2fV24-H2H-Hτ1fVτ1I2H2fV2I-Iτ2D2ID-DV2D2V+fVfV2-VV2+ln⁡Hτ1fVτ1Iτ2HfVI=μH21-H2H1-HH2+kH2fV2-φH2H-φHτ1fVτ1I2H2fV2I-φIτ2D2ID-φDV2D2V+fVfV2-VV2+ln⁡fV2VfVV2=μH21-H2H1-HH2+kH2fV2-φH2H-φHτ1fVτ1I2H2fV2I-φIτ2D2ID-φDV2D2V-φfV2VfVV2+1-fV2fVfVfV2-VV2.Similar to the proof of Theorem 6, we have dL3/dt≤0 for all H,I,D,V,Z>0. Furthermore, it can be shown that the largest invariant subset of dL3/dt=0 is the singleton {E2}. Hence, the global asymptotic stability of the infection equilibrium with immunity E2 follows from LaSalle’s invariant principle. This completes the proof.


## 2.1. Preliminaries

The following result established the positivity and boundedness of solutions of model (2).Theorem 1.
Let(H(t),I(t),D(t),V(t),Z(t)) be the solution of model (2) satisfying conditions (5). Then H(t),I(t),D(t),V(t) and Z(t) are all nonnegative and bounded for all t>0.Proof.
First, we prove thatH(t)>0 for all t≥0. Assume the contrary and let T0>0 such that H(T0)=0. Then from the first equation of system (2), we have H′(t)=λ>0. Therefore, H(t)<0 for t∈(T0-ε,T0) and ε>0 is sufficiently small. This contradicts with the fact of H(t)>0 for t∈[0,T0). It follows that H(t)>0 for t≥0. Moreover, it follows from (2) that (8)It=I0e-∫0tδ+pZθdθ+∫0te-∫θtδ+pZθdθkHθ-τ1fVθ-τ1dθ,Dt=D0e-β+δt+∫0taIθ-τ2eβ+δθ-tdθ,Vt=V0e-ct+∫0tβDθecθ-tdθ,Zt=Z0e-∫0tσ-qIθdθ.Then the nonnegative immediately follows from the above integral forms and (5).
Next, we show the boundedness of the solution. Define(9)Gt=Ht+It+τ1+δ2aDt+τ1+τ2+δ2aVt+τ1+τ2+pqZt+τ1,and d0=minμ,δ/2,c,σ. It then follows that (10)G′t=s-μHt-δ2It+τ1-δ22Dt+τ1+τ2-δc2aVt+τ1+τ2-σpqZt+τ1≤s-d0Gt.This implies that G(t) is bounded and so are H(t),I(t),D(t),V(t), and Z(t). This completes the proof.


## 2.2. Steady States

Obviously, the model (2) always has an infection-free equilibrium E0=(H0,0,0,0,0) with H0=s/μ. This is the only biologically meaningful equilibrium if (11)R0=aksβf′0δcμβ+δ<1.At an equilibrium of model (2) we have (12)s=μH+kHfV,kHfV=δI+pIZ,aI=β+δD,βD=cV,qI-σZ=0.It is clear that model (2) only has the following two possible equilibria except the E0, that is, E1=(H1,I1,D1,V1,0) and E2=(H2,I2,D2,V2,Z2), where H1,I1,D1,V1,H2,I2,D2,V2,Z2 are all strictly positive.IfZ=0, then the existence of E1=(H1,I1,D1,V1,0) is equivalent to the existence of positive solution (H1,I1,D1,V1) of the following equations: (13)s=μH+kHfV,kHfV=δI,aI=β+δD,βD=cV.From the last three equations of (13) we have (14)s-μH=δcβ+δVaβ,I=β+δcVaβ,D=cVβ.This means that, in order to have H≥0 and V>0 at an equilibrium, we must have V∈(0,aβs/δc(β+δ)]. Further, substituting I into the second equation of (13) gives (15)H=δcβ+δVakβfV.Then substituting H into the first equation of (13), direct calculation yields (16)0=s-μδcβ+δVaβkfV-δcβ+δVaβ≕FV.For F(V) with V>0, it then follows from (4) that (17)F′V=-δcβ+δaβ-μδcβ+δakβfV-f′VV<0.Moreover, we obtain that (18)limV→0+FV=μδcβ+δaβkf′0R0-1,Faβsδcβ+δ=-μskfaβs/δcβ+δ<0.Therefore, there exists an infection equilibrium without immunity E1=(H1,I1,D1,V1,0) when R0>1.Define(19)R1=qksfβaσ/qcβ+δδσμ+kfβaσ/qcβ+δwhich represents the immune response activation number and determines whether or not a persistent immune response can be established. If Z≠0, then from (12) we have (20)H2=sμ+kfV2,I2=σq,D2=aσqβ+δ,V2=aβσqcβ+δ,Z2=δpR1-1.Therefore, the infection equilibrium with immunity E2=(H2,I2,D2,V2,Z2) exists provided R1>0. It follows from the properties of function f(V) that R1<R0. Concluding the above analysis we have the following result.Theorem 2.
For model (2).(i)
IfR0<1, then there exists a unique infection-free equilibrium E0;(ii)
IfR1≤1<R0, then there exists a unique infection equilibrium without immunity E1 besides E0;(iii)
IfR1>1, then there exists a unique infection equilibrium with immunity E2 besides E0 and E1.


## 2.3. Global Dynamics of Model (2)

In this part, we investigate the global asymptotic stability of the equilibria of system (2) by constructing Lyapunov functionals. To this end, we first introduce the following function for the following work, φx=x-1-ln⁡x for x∈(0,+∞), which will be used in Lyapunov functionals. It is easy to show that φ(x)≥φ(1)=0.Theorem 3.
IfR0≤1, then the infection-free equilibrium E0 is globally asymptotically stable.Proof.
We construct a Lyapunov functional as follows:(21)L1=H0φHH0+I+δaD+δβ+δaβV+pqZ+∫t-τ1tkHθfVθdθ+∫t-τ2tIθdθ.To proceed, we denote uτ=u(t-τ) (u∈{H,I,V}, τ∈{τ1,τ2}), for the sake of convenience. Calculating dL1/dt along the solutions of system (2) and applying s=μH0, together with condition (4), yield (22)dL1dt=1-H0Hs-μH-kHfV+kHτ1fVτ1-δI-pIZ+δaaIτ2-β+δD+δβ+δaββD-cV+kHfV-Hτ1fVτ1+δI-Iτ2=μH01-H0H1-HH0+kH0fV-δβ+δcVaβ-pσqZ≤μH01-H0H1-HH0+δcβ+δaβR0-1V-pσqZ.Therefore, if R0≤1, then dL1/dt≤0. Furthermore, it can be shown that the largest invariant subset of dL1/dt=0 is the singleton {E0}. Thus, the infection-free equilibrium E0 is globally asymptotically stable. This completes the proof.To establish the global stability of the infection equilibrium without immunityE1 when R1≤1<R0, we first give the following results.Lemma 4.
Under condition (3), for Vi>0,i=1,2, it holds that (23)fV1-fV2V1-V2>0,(24)fV2V2-fV1V1V1-V2>0,(25)fVfVi-VVi1-fVifV≤0,i=1,2.This Lemma can be easily obtained from the properties of functionf(V). Hence, we omit the proof. Based on Lemma 4, we present the following Lemma.Lemma 5.
Suppose the condition (3) is satisfied and R0>1. Then Hi,Ii,Di,Vi(i=1,2) exist satisfying (26)sgnH2-H1=sgnI1-I2=sgnD1-D2=sgnV1-V2=sgnR1-1.Proof.
First, we claim thatsgn(H2-H1)=sgn(I1-I2). On the contrary, we have sgn(H1-H2)=sgn(I1-I2). It follows from the conditions of the equilibria E1 and E2 that sgn(I1-I2)=sgn(D1-D2)=sgn(V1-V2); furthermore, (27)μH2-H1=kH1fV1-H2fV2=kH1-H2fV1+kH2fV1-fV2.Thus, from (23) in Lemma 4 we get sgn(H2-H1)=sign(H1-H2) which leads to a contradiction. Thus, sign(H2-H1)=sgn(I1-I2).
Next, we will provesgn(R1-1)=sgn(I1-I2). To this end, let (28)R~1=kH2fV2δI2;then it is easy to see that sgn(R1-1)=sgn(R~1-1). Thus, we just need to prove sgn(R~1-1)=sgn(I1-I2). Use the equilibria conditions of E1 and E2, which gives (29)R~-1=kH2fV2δI2-kH1fV1δI1=akβδcβ+δH2fV2V2-H1fV1V1=akβδcβ+δH2-H1fV2V2+H1fV2V2-fV1V1.Thus, it follows from (24) that sgn(R~1-1)=sgn(I1-I2). This completes the proof.Theorem 6.
IfR1≤1<R0, then the infection equilibrium without immunity E1 is globally asymptotically stable.Proof.
Constructing a Lyapunov functionalL2 as follows(30)L2=H1φHH1+I1φII1+δaD1φDD1+δβ+δV1aβφVV1+pqZ+kH1fV1∫t-τ1tφHθfVθH1fV1dθ+δI1∫t-τ2tφIθI1dθ.Take the derivative of L2 along solutions of model (2) and recall that the equilibrium conditions of E1 are(31)s=μH1+kH1fV1,kH1fV1=δI1=δβ+δD1a=δcβ+δV1aβ.Then, we obtain that (32)dL2dt=1-H1Hs-μH-kHfV+1-I1IkHτ1fVτ1-δI-pIZ+δa1-D1DaIτ2-β+δD+δβ+δaβ1-V1VβD-cV+pqqIZ-σZ+kH1fV1HfVH1fV1-Hτ1fVτ1H1fV1+ln⁡Hτ1fVτ1HfV+δI1II1-Iτ2I1+ln⁡Iτ2I=μH11-H1H1-HH1+1-H1HkH1fV1-kHfV+1-I1IkHτ1fVτ1-δI1II1+δa1-D1DaI1Iτ2I1-β+δδD1aDD1+δβ+δaβ1-V1VβD1DD1-cV1VV1+pZI1-σq+kH1fV1HfVH1fV1-Hτ1fVτ1H1fV1+ln⁡Hτ1fVτ1HfV+δI1II1-Iτ2I1+ln⁡Iτ2I=μH11-H1H1-HH1+kH1fV14-H1H-Hτ1fVτ1I1H1fV1I-Iτ2D1ID-DV1D1V+fVfV1-VV1+ln⁡Hτ1fVτ1Iτ2HfVI+pZI1-I2=μH11-H1H1-HH1+kH1fV1-φH1H-φHτ1fVτ1I1H1fV1I-φIτ2D1ID-φDV1D1V+fVfV1-VV1+ln⁡fV1VfVV1+pZI1-I2=μH11-H1H1-HH1+kH1fV1-φH1H-φHτ1fVτ1I1H1fV1I-φIτ2D1ID-φDV1D1V-φfV1VfVV1+1-fV1fVfVfV1-VV1+pZI1-I2.From Lemma 5 we can obtain that R1≤1 is equivalent to I1-I2≤0. Combined with (25) in Lemma 4, it follows that dL2/dt≤0 for all H,I,D,V,Z>0. Furthermore, it can be shown that the largest invariant subset of dL2/dt=0 is the singleton {E1}. Hence, the global asymptotic stability of the infection equilibrium without immunity E1 follows from LaSalle’s invariant principle. This completes the proof.Theorem 7.
IfR1>1, then the infection equilibrium with immunity E2 is globally asymptotically stable.Proof.
Constructing a Lyapunov functionalL3 as follows: (33)L3=H2φHH2+I2φII2+δ+pZ2aD2φDD2+δ+pZ2β+δV2aβφVV2+pqφZZ2+kH2fV2∫t-τ1tφHθfVθH2fV2dθ+δ+pZ2I2∫t-τ2tφIθI2dθ.Take the derivative of L3 along solutions of model (2) and recall that the equilibrium conditions of E2 are (34)s=μ+kfV2H2,I2=σq,kH2fV2=δ+pZ2I2=δ+pZ2β+δD2a=δ+pZ2β+δcV2aβ.Then, we obtain that (35)dL3dt=1-H2Hs-μH-kHfV+1-I2IkHτ1fVτ1-δI-pIZ+δ+pZ2a1-D2DaIτ2-β+δD+δ+pZ2β+δaβ1-V2VβD-cV+pq1-Z2ZqIZ-σZ+kH2fV2HfVH2fV2-Hτ1fVτ1H2fV2+ln⁡Hτ1fVτ1HfV+δ+pZ2I2II2-Iτ2I2+ln⁡Iτ2I=μH21-H2H1-HH2+1-H2HkH2fV2-kHfV+1-I2IkHτ1fVτ1-kH2fV2II2+pIZ2-pIZ+δ+pZ2a1-D2DaI2Iτ2I2-β+δδD2aDD2+δ+pZ2β+δaβ1-V2VβD1DD1-cV1VV1+pq1-Z2ZqIZ-σZ+kH2fV2HfVH2fV2-Hτ1fVτ1H2fV2+ln⁡Hτ1fVτ1HfV+δ+pZ2I2II2-Iτ2I2+ln⁡Iτ2I=μH21-H2H1-HH2+kH2fV24-H2H-Hτ1fVτ1I2H2fV2I-Iτ2D2ID-DV2D2V+fVfV2-VV2+ln⁡Hτ1fVτ1Iτ2HfVI=μH21-H2H1-HH2+kH2fV2-φH2H-φHτ1fVτ1I2H2fV2I-φIτ2D2ID-φDV2D2V+fVfV2-VV2+ln⁡fV2VfVV2=μH21-H2H1-HH2+kH2fV2-φH2H-φHτ1fVτ1I2H2fV2I-φIτ2D2ID-φDV2D2V-φfV2VfVV2+1-fV2fVfVfV2-VV2.Similar to the proof of Theorem 6, we have dL3/dt≤0 for all H,I,D,V,Z>0. Furthermore, it can be shown that the largest invariant subset of dL3/dt=0 is the singleton {E2}. Hence, the global asymptotic stability of the infection equilibrium with immunity E2 follows from LaSalle’s invariant principle. This completes the proof.


## 3. Global Dynamics of the Discrete Model (6)

In this section, we will show that the discrete model (6) can efficiently preserve the global asymptotic stability of the equilibria for corresponding continuous model (2).It is easy to validate that model (6) has the same equilibria as model (2). We also denote the equilibria as E0=(x0,0,0,0) and E1=(H1,I1,D1,V1,0) and E2=(H2,I2,D2,V2,Z2).The following equations can be easily obtained by rearranging the formulations in equations of (6): (36)Hn+1=sΔt+Hn1+Δtμ+kfVn,In+1=In+ΔtkHn-m1+1fVn-m11+Δtδ+pZn+1,Dn+1=Dn+ΔtaIn-m2+11+Δtβ+δ,Vn+1=Vn+ΔtβDn+11+Δtc,Zn+1=Zn1+Δtσ-qIn+1.Lemma 8.
The solution(Hn,In,Dn,Vn,Zn) of system (6) subject to condition (7) exists uniquely and is positive and bounded for all n∈N. In addition, 0<In<1+Δtσ/Δtq for n=1,2,….Proof.
We first validate that(H1,I1,D1,V1,Z1) exists uniquely and is positive. According to the first equation (36), we know that H1>0. Next, we consider Z1. It follows from the second and fifth equations of (36) that (37)Z1=Z0+ΔtqI0+ΔtkH-m1+1fV-m11+Δtδ+pZ1Z1-σZ1.Define (38)hZ1=Δtp1+ΔtσZ12+1+Δtδ1+Δtσ-ΔtpZ0-ΔtqI0+ΔtkH-m1+1fV-m1Z1-1+ΔtδZ0,which is a quadratic function. Note that h(0)=-(1+Δtδ)Z0<0 and limZ1→∞h(Z1)=∞; there is a unique Z1>0 such that h(Z1)=0. That is, (37) holds.
Furthermore, we considerI1. Combining with the second and last equation of (36), we have (39)I1=I0+ΔtkH-m1+1fV-m1-δI1-pI1Z01+Δtσ-qI1.Define (40)hI1=Δtq1+ΔtδI12-1+Δtδ1+Δtσ+ΔtpZ0+ΔtqI0+ΔtkH-m1+1fV-m1I1+1+ΔtσI0+ΔtkH-m1+1fV-m1.Since Z1>0, it follows from the last equation of system (36) that I1<1+Δtσ/Δtq. Then we have (41)h0=1+ΔtσI0+ΔtkH-m1+1fV-m1>0,h1+ΔtσΔtq=-pZ01+Δtσq<0.Due to h(I1) being a quadratic function, there exists a unique I1∈0,1+Δtσ/Δtq such that h(I1)=0. Therefore, (39) holds. Finally, we consider D1 and V1. It follows from the third and fourth equations of (36) that D1=D0+ΔtaI-m2+1/1+Δt(β+σ), V1=V0+ΔtβD1/1+Δtc. Thus, D1 and V1 uniquely exist and are positive. Therefore, (H1,I,D1,V1,Z1) exists uniquely and is positive.
Forn=1, repeat the above process; we can show that (H2,I2,D2,V2,Z2) exists uniquely and is positive. Owing to Z2>0, we also have I2<1+Δtσ/Δtq. Therefore, by using the mathematical induction, for all n≥0, we know that (Hn,In,Dn,Vn,Zn) exists uniquely and is positive with In<1+Δtσ/Δtq. This completes the proof.
In order to prove the boundedness of solutions, we define(42)Un=Hn+In+m1+δ2aDn+m1+m2+Vn+m1+m2+pqZn+m1.Then we have (43)Un+1-Un=Δts-μHn+1-δ2aIn+m1+1-δ22aDn+m1+m2+1-δc2aVn+m1+m2+1-pσqZn+m1≤Δts-d0Un+1,where d0=min⁡μ,δ/2,c,σ. Thus, we obtain (44)Un+1≤11+Δtd0Un+Δts1+Δtd0.The following inequality can be easily reduced from the induction (45)Un≤11+Δtd0nU0+sd01-11+Δd0n,which implies that (46)limn→∞supUn≤sd0.This means that {Un} is bounded. Therefore, {Hn},{In},{Dn},{Vn} and {Zn} are bounded. This completes the proof.


### 3.1. Global Stability of Equilibria

Theorem 9.
IfR0≤1, then the infection-free equilibrium E0 of system (6) is globally asymptotically stable.Proof.
Define a discrete Lyapunov function(47)Wn1=1ΔtH0φHnH0+In+δaDn+δβ+δaβ1+ΔtcVn+pq1+ΔtσZn+∑j=n-m1n-1kHj+1fVj+δ∑j=n-m2n-1Ij+1.Recall that s=μH0; then we have (48)Wn+11-Wn1=1ΔtHn+1-Hn+H0lnHnHn+1+In+1-In+δaDn+1-Dn+δβ+δaβ1+ΔtcVn+1-Vn+pq1+ΔtσZn+1-Zn+kHn+1fVn-kHn-m1+1fVn-m1+δIn+1-δIn-m2+1≤1Δt1-H0Hn+1Hn+1-Hn+In+1-In+δaDn+1-Dn+δβ+δaβ1+ΔtcVn+1-Vn+pq1+ΔtσZn+1-Zn+kHn+1fVn-kHn-m1+1fVn-m1+δIn+1-δIn-m2+1=1-H0Hn+1μH0-μHn+1-kHn+1fVn+kHn-m1+1fVn-m1-δIn+1+δaaIn-m2+1-β+δDn+1+δβ+δaββDn+1-cVn+1+δcβ+δaβVn+1-Vn+pqqIn+1Zn+1-σZn+1+pσqZn+1-Zn+kHn+1fVn-kHn-m1+1fVn-m1+δIn+1-δIn-m2+1=μH01-H0Hn+11-Hn+1H0+kH0fVn-δcβ+δaβVn-pσqZn≤μH01-H0Hn+11-Hn+1H0+δcβ+δaβVnR0-1-pσqZn.It then follows that if R0≤1, Wn+1(1)-Wn(1)≤0, for all n∈N. Thus, Wn(1) is monotone decreasing sequence. Due to Wn(1)≥0, there is a limit limn→∞Wn(1)≥0 which implies that limn→∞(Wn+1(1)-Wn(1))=0. Moreover, it can be shown that limn→∞Hn=H0, limn→∞In=0, limn→∞Dn=0, limn→∞Vn=0, limn→∞Zn=0. Hence, E0 is globally asymptotically. This completes the proof.Theorem 10.
IfR1≤1<R0, then the infection equilibrium without immunity E1 of system (6) is globally asymptotically stable.Proof.
Define a discrete Lyapunov functional as follows:(49)Wn2=1ΔtH1φHnH1+I1φInI1+δaD1φDnD1+δβ+δaβV1φVnV1+pqZn+kH1fV1∑j=n-m1n-1φHj+1fVjH1fV1+δI1∑j=n-m2n-1φIj+1I1.Combine with the model (6) and the equilibrium conditions (31) for E1. Then we get the difference of Wn(2)(50)Wn+12-Wn2=1ΔtHn+1-Hn+H1ln⁡HnHn+1+In+1-In+I1ln⁡InIn+1+δaDn+1-Dn+D1ln⁡DnDn+1+δβ+δaβVn+1-Vn+V1lnVnVn+1+pqZn+1-Zn+kH1fV1∑j=n-m1+1nφHj+1fVjH1fV1-∑j=n-m1n-1φHj+1fVjH1fV1+δI1∑j=n-m2+1nφIj+1I1-∑j=n-m2n-1φIj+1I1+kH1fV1φfVn+1fV1-φfVnfV1≤1Δt1-H1Hn+1Hn+1-Hn+1-I1In+1In+1-In+δa1-D1Dn+1×Dn+1-Dn+δβ+δaβ1-V1Vn+1Vn+1-Vn+pqZn+1-Zn+kH1fV1∑j=n-m1+1nφHj+1fVjH1fV1-∑j=n-m1n-1φHj+1fVjH1fV1+δI1∑j=n-m2+1nφIj+1I1-∑j=n-m2n-1φIj+1I1+kH1fV1φfVn+1fV1-φfVnfV1=1-H1Hn+1s-μHn+1-kHn+1fVn+1-I1In+1kHn-m1+1fVn-m1-δIn+1-pIn+1Zn+1+δa1-D1Dn+1aIn-m2+1-β+δDn+1+δβ+δaβ1-V1Vn+1×βDn+1-cVn+1+pqZn+1-Zn+kH1fV1φHn+1fVnH1fV1-φHn-m1+1fVn-m1H1fV1+δI1φIn+1I1-φIn-m2+1I1+kH1fV1φfVn+1fV1-φfVnfV1=μH11-H1Hn+11-Hn+1H1+1-H1Hn+1kH1fV1-kHn+1fVn+1-I1In+1kHn-m1+1fVn-m1-δI1In+1I1+δa1-D1Dn+1×aI1In-m2+1I1-β+δD1Dn+1D1+δβ+δaβ1-V1Vn+1×βD1Dn+1D1-cV1Vn+1V1+kH1fV1Hn+1fVnH1fV1-Hn-m1+1fVn-m1H1fV1+ln⁡Hn-m1+1fVn-m1Hn+1fVn+δI1In+1I1-In-m2+1I1+lnIn-m2+1I1+kH1fV1fVn+1fV1-fVnfV1+ln⁡fVnfVn+1+pZn+1I1-σq=μH11-H1Hn+11-Hn+1H1+kH1fV14-H1Hn+1-I1Hn-m1+1fVn-m1In+1H1fV1-D1In-m2+1Dn+1I1-V1Dn+1Vn+1D1-Vn+1V1+fVnfV1+ln⁡Hn-m1+1fVn-m1In-m2+1Hn+1In+1fVn+1+pZn+1I1-I2=μH11-H1Hn+11-Hn+1H1+kH1fV1-φH1Hn+1-φD1In-m2+1Dn+1I1-φI1Hn-m1+1fVn-m1In+1H1fV1-φV1Dn+1Vn+1D1+fVn+1fV1-Vn+1V1+ln⁡fV1Vn+1fVn+1V1+pZn+1I1-I2=μH11-H1Hn+11-Hn+1H1+kH1fV1-φH1Hn+1-φD1In-m2+1Dn+1I1-φI1Hn-m1+1fVn-m1In+1H1fV1-φV1Dn+1Vn+1D1-φfV1Vn+1V1fVn+1+1-fV1fVn+1fVn+1fV1-Vn+1V1+pZn+1I1-I2.Similar to the proof of Theorem 6, we get Wn+1(2)-Wn(2)≤0, for all n∈N. That is, Wn(2) is monotone decreasing sequence. Furthermore, since Wn(2)>0, there is a limit limn→∞Wn(2)≥0. Hence, limn→∞(Wn+1(2)-Wn(2))=0. Furthermore, from model (6), it is easy to show that limn→∞Hn=H1, limn→∞In=H1, limn→∞Dn=D1, limn→∞Vn=V1, limn→∞Zn=0, which implies that E1 is globally asymptotically stable. This completes the proof.Theorem 11.
IfR1>1, then the infection equilibrium with immunity E2 is globally asymptotically stable.Proof.
Define a discrete Lyapunov functional as follows:(51)Wn3=1ΔtH2φHnH2+I2φInI2+δaD2φDnD2+δβ+δaβV2φVnV2+pqφZnZ2+kH2fV2∑j=n-m1n-1φHj+1fVjH2fV2+δI2∑j=n-m2n-1φIj+1I2.Combine with the model (6) and the equilibrium conditions (34) for E2; then we obtain the difference of Wn(3)(52)Wn+13-Wn3=1ΔtHn+1-Hn+H2ln⁡HnHn+1+In+1-In+I2ln⁡InIn+1+δaDn+1-Dn+D2ln⁡DnDn+1+δβ+δaβVn+1-Vn+V2ln⁡VnVn+1+pqZn+1-Zn+ln⁡ZnZn+1+kH2fV2∑j=n-m1+1nφHj+1fVjH2fV2-∑j=n-m1n-1φHj+1fVjH2fV2+δI2∑j=n-m2+1nφIj+1I2-∑j=n-m2n-1φIj+1I2+kH2fV2φfVn+1fV2-φfVnfV2≤1Δt1-H2Hn+1Hn+1-Hn+1-I2In+1In+1-In+δa1-D2Dn+1×Dn+1-Dn+δβ+δaβ1-V2Vn+1Vn+1-Vn+pq1-Z2Zn+1Zn+1-Zn+kH2fV2∑j=n-m1+1nφHj+1fVjH2fV2-∑j=n-m1n-1φHj+1fVjH2fV2+δI2∑j=n-m2+1nφIj+1I2-∑j=n-m2n-1φIj+1I2+kH2fV2φfVn+1fV2-φfVnfV2=1-H2Hn+1s-μHn+1-kHn+1fVn+1-I2In+1kHn-m1+1fVn-m1-δIn+1-pIn+1Zn+1+δa1-D2Dn+1aIn-m2+1-β+δDn+1+δβ+δaβ1-V2Vn+1βDn+1-cVn+1+pq1-Z2Zn+1qIn+1Zn+1-σZn+kH2fV2φHn+1fVnH2fV2-φHn-m1+1fVn-m1H2fV2+δI2φIn+1I2-φIn-m2+1I2+kH2fV2φfVn+1fV2-φfVnfV2=μH21-H2Hn+11-Hn+1H2+1-H2Hn+1kH2fV2-kHn+1fVn+1-I2In+1kHn-m1+1fVn-m1-kH2fV2In+1I2+pIn+1Z2-pIn+1Zn+1+δa1-D2Dn+1aI2In-m2+1I2-β+δD2Dn+1D2+δβ+δaβ1-V2Vn+1×βD2Dn+1D2-cV2Vn+1V2+kH2fV2Hn+1fVnH2fV2-Hn-m1+1fVn-m1H2fV2+ln⁡Hn-m1+1fVn-m1Hn+1fVn+δI2In+1I2-In-m2+1I2+ln⁡In-m2+1I2+kH2fV2fVn+1fV2-fVnfV2+ln⁡fVnfVn+1=μH21-H2Hn+11-Hn+1H2+kH2fV24-H2Hn+1-I2Hn-m1+1fVn-m1In+1H2fV2-D2In-m2+1Dn+1I2-V2Dn+1Vn+1D2-Vn+1V2+fVnfV2+lnHn-m1+1fVn-m1In-m2+1Hn+1In+1fVn+1=μH21-H2Hn+11-Hn+1H2+kH2fV2-φH2Hn+1-φD2In-m2+1Dn+1I2-φI2Hn-m1+1fVn-m1In+1H2fV2-φV2Dn+1Vn+1D2+fVn+1fV2-Vn+1V2+ln⁡fV2Vn+1fVn+1V2=μH21-H2Hn+11-Hn+1H2+kH2fV2-φH2Hn+1-φD2In-m2+1Dn+1I2-φI2Hn-m1+1fVn-m1In+1H1fV2-φV2Dn+1Vn+1D2-φfV2Vn+1V2fVn+1+1-fV2fVn+1fVn+1fV2-Vn+1V2.Similar to the proof of Theorem 10, if R1>1, then we have Wn+1(3)-Wn(3)≤0, for all n∈N. That is, Wn(3) is monotone decreasing sequence. Furthermore, since Wn(3)>0, there is a limit limn→∞Wn(3)≥0. Hence, limn→∞(Wn+1(3)-Wn(3))=0. Combined with model (6), it can be shown that limn→∞Hn=H2, limn→∞In=I2, limn→∞Dn=D2, limn→∞Vn=V2, limn→∞Zn=Z2, which implies that E2 is globally asymptotically stable. This completes the proof.


## 3.1. Global Stability of Equilibria

Theorem 9.
IfR0≤1, then the infection-free equilibrium E0 of system (6) is globally asymptotically stable.Proof.
Define a discrete Lyapunov function(47)Wn1=1ΔtH0φHnH0+In+δaDn+δβ+δaβ1+ΔtcVn+pq1+ΔtσZn+∑j=n-m1n-1kHj+1fVj+δ∑j=n-m2n-1Ij+1.Recall that s=μH0; then we have (48)Wn+11-Wn1=1ΔtHn+1-Hn+H0lnHnHn+1+In+1-In+δaDn+1-Dn+δβ+δaβ1+ΔtcVn+1-Vn+pq1+ΔtσZn+1-Zn+kHn+1fVn-kHn-m1+1fVn-m1+δIn+1-δIn-m2+1≤1Δt1-H0Hn+1Hn+1-Hn+In+1-In+δaDn+1-Dn+δβ+δaβ1+ΔtcVn+1-Vn+pq1+ΔtσZn+1-Zn+kHn+1fVn-kHn-m1+1fVn-m1+δIn+1-δIn-m2+1=1-H0Hn+1μH0-μHn+1-kHn+1fVn+kHn-m1+1fVn-m1-δIn+1+δaaIn-m2+1-β+δDn+1+δβ+δaββDn+1-cVn+1+δcβ+δaβVn+1-Vn+pqqIn+1Zn+1-σZn+1+pσqZn+1-Zn+kHn+1fVn-kHn-m1+1fVn-m1+δIn+1-δIn-m2+1=μH01-H0Hn+11-Hn+1H0+kH0fVn-δcβ+δaβVn-pσqZn≤μH01-H0Hn+11-Hn+1H0+δcβ+δaβVnR0-1-pσqZn.It then follows that if R0≤1, Wn+1(1)-Wn(1)≤0, for all n∈N. Thus, Wn(1) is monotone decreasing sequence. Due to Wn(1)≥0, there is a limit limn→∞Wn(1)≥0 which implies that limn→∞(Wn+1(1)-Wn(1))=0. Moreover, it can be shown that limn→∞Hn=H0, limn→∞In=0, limn→∞Dn=0, limn→∞Vn=0, limn→∞Zn=0. Hence, E0 is globally asymptotically. This completes the proof.Theorem 10.
IfR1≤1<R0, then the infection equilibrium without immunity E1 of system (6) is globally asymptotically stable.Proof.
Define a discrete Lyapunov functional as follows:(49)Wn2=1ΔtH1φHnH1+I1φInI1+δaD1φDnD1+δβ+δaβV1φVnV1+pqZn+kH1fV1∑j=n-m1n-1φHj+1fVjH1fV1+δI1∑j=n-m2n-1φIj+1I1.Combine with the model (6) and the equilibrium conditions (31) for E1. Then we get the difference of Wn(2)(50)Wn+12-Wn2=1ΔtHn+1-Hn+H1ln⁡HnHn+1+In+1-In+I1ln⁡InIn+1+δaDn+1-Dn+D1ln⁡DnDn+1+δβ+δaβVn+1-Vn+V1lnVnVn+1+pqZn+1-Zn+kH1fV1∑j=n-m1+1nφHj+1fVjH1fV1-∑j=n-m1n-1φHj+1fVjH1fV1+δI1∑j=n-m2+1nφIj+1I1-∑j=n-m2n-1φIj+1I1+kH1fV1φfVn+1fV1-φfVnfV1≤1Δt1-H1Hn+1Hn+1-Hn+1-I1In+1In+1-In+δa1-D1Dn+1×Dn+1-Dn+δβ+δaβ1-V1Vn+1Vn+1-Vn+pqZn+1-Zn+kH1fV1∑j=n-m1+1nφHj+1fVjH1fV1-∑j=n-m1n-1φHj+1fVjH1fV1+δI1∑j=n-m2+1nφIj+1I1-∑j=n-m2n-1φIj+1I1+kH1fV1φfVn+1fV1-φfVnfV1=1-H1Hn+1s-μHn+1-kHn+1fVn+1-I1In+1kHn-m1+1fVn-m1-δIn+1-pIn+1Zn+1+δa1-D1Dn+1aIn-m2+1-β+δDn+1+δβ+δaβ1-V1Vn+1×βDn+1-cVn+1+pqZn+1-Zn+kH1fV1φHn+1fVnH1fV1-φHn-m1+1fVn-m1H1fV1+δI1φIn+1I1-φIn-m2+1I1+kH1fV1φfVn+1fV1-φfVnfV1=μH11-H1Hn+11-Hn+1H1+1-H1Hn+1kH1fV1-kHn+1fVn+1-I1In+1kHn-m1+1fVn-m1-δI1In+1I1+δa1-D1Dn+1×aI1In-m2+1I1-β+δD1Dn+1D1+δβ+δaβ1-V1Vn+1×βD1Dn+1D1-cV1Vn+1V1+kH1fV1Hn+1fVnH1fV1-Hn-m1+1fVn-m1H1fV1+ln⁡Hn-m1+1fVn-m1Hn+1fVn+δI1In+1I1-In-m2+1I1+lnIn-m2+1I1+kH1fV1fVn+1fV1-fVnfV1+ln⁡fVnfVn+1+pZn+1I1-σq=μH11-H1Hn+11-Hn+1H1+kH1fV14-H1Hn+1-I1Hn-m1+1fVn-m1In+1H1fV1-D1In-m2+1Dn+1I1-V1Dn+1Vn+1D1-Vn+1V1+fVnfV1+ln⁡Hn-m1+1fVn-m1In-m2+1Hn+1In+1fVn+1+pZn+1I1-I2=μH11-H1Hn+11-Hn+1H1+kH1fV1-φH1Hn+1-φD1In-m2+1Dn+1I1-φI1Hn-m1+1fVn-m1In+1H1fV1-φV1Dn+1Vn+1D1+fVn+1fV1-Vn+1V1+ln⁡fV1Vn+1fVn+1V1+pZn+1I1-I2=μH11-H1Hn+11-Hn+1H1+kH1fV1-φH1Hn+1-φD1In-m2+1Dn+1I1-φI1Hn-m1+1fVn-m1In+1H1fV1-φV1Dn+1Vn+1D1-φfV1Vn+1V1fVn+1+1-fV1fVn+1fVn+1fV1-Vn+1V1+pZn+1I1-I2.Similar to the proof of Theorem 6, we get Wn+1(2)-Wn(2)≤0, for all n∈N. That is, Wn(2) is monotone decreasing sequence. Furthermore, since Wn(2)>0, there is a limit limn→∞Wn(2)≥0. Hence, limn→∞(Wn+1(2)-Wn(2))=0. Furthermore, from model (6), it is easy to show that limn→∞Hn=H1, limn→∞In=H1, limn→∞Dn=D1, limn→∞Vn=V1, limn→∞Zn=0, which implies that E1 is globally asymptotically stable. This completes the proof.Theorem 11.
IfR1>1, then the infection equilibrium with immunity E2 is globally asymptotically stable.Proof.
Define a discrete Lyapunov functional as follows:(51)Wn3=1ΔtH2φHnH2+I2φInI2+δaD2φDnD2+δβ+δaβV2φVnV2+pqφZnZ2+kH2fV2∑j=n-m1n-1φHj+1fVjH2fV2+δI2∑j=n-m2n-1φIj+1I2.Combine with the model (6) and the equilibrium conditions (34) for E2; then we obtain the difference of Wn(3)(52)Wn+13-Wn3=1ΔtHn+1-Hn+H2ln⁡HnHn+1+In+1-In+I2ln⁡InIn+1+δaDn+1-Dn+D2ln⁡DnDn+1+δβ+δaβVn+1-Vn+V2ln⁡VnVn+1+pqZn+1-Zn+ln⁡ZnZn+1+kH2fV2∑j=n-m1+1nφHj+1fVjH2fV2-∑j=n-m1n-1φHj+1fVjH2fV2+δI2∑j=n-m2+1nφIj+1I2-∑j=n-m2n-1φIj+1I2+kH2fV2φfVn+1fV2-φfVnfV2≤1Δt1-H2Hn+1Hn+1-Hn+1-I2In+1In+1-In+δa1-D2Dn+1×Dn+1-Dn+δβ+δaβ1-V2Vn+1Vn+1-Vn+pq1-Z2Zn+1Zn+1-Zn+kH2fV2∑j=n-m1+1nφHj+1fVjH2fV2-∑j=n-m1n-1φHj+1fVjH2fV2+δI2∑j=n-m2+1nφIj+1I2-∑j=n-m2n-1φIj+1I2+kH2fV2φfVn+1fV2-φfVnfV2=1-H2Hn+1s-μHn+1-kHn+1fVn+1-I2In+1kHn-m1+1fVn-m1-δIn+1-pIn+1Zn+1+δa1-D2Dn+1aIn-m2+1-β+δDn+1+δβ+δaβ1-V2Vn+1βDn+1-cVn+1+pq1-Z2Zn+1qIn+1Zn+1-σZn+kH2fV2φHn+1fVnH2fV2-φHn-m1+1fVn-m1H2fV2+δI2φIn+1I2-φIn-m2+1I2+kH2fV2φfVn+1fV2-φfVnfV2=μH21-H2Hn+11-Hn+1H2+1-H2Hn+1kH2fV2-kHn+1fVn+1-I2In+1kHn-m1+1fVn-m1-kH2fV2In+1I2+pIn+1Z2-pIn+1Zn+1+δa1-D2Dn+1aI2In-m2+1I2-β+δD2Dn+1D2+δβ+δaβ1-V2Vn+1×βD2Dn+1D2-cV2Vn+1V2+kH2fV2Hn+1fVnH2fV2-Hn-m1+1fVn-m1H2fV2+ln⁡Hn-m1+1fVn-m1Hn+1fVn+δI2In+1I2-In-m2+1I2+ln⁡In-m2+1I2+kH2fV2fVn+1fV2-fVnfV2+ln⁡fVnfVn+1=μH21-H2Hn+11-Hn+1H2+kH2fV24-H2Hn+1-I2Hn-m1+1fVn-m1In+1H2fV2-D2In-m2+1Dn+1I2-V2Dn+1Vn+1D2-Vn+1V2+fVnfV2+lnHn-m1+1fVn-m1In-m2+1Hn+1In+1fVn+1=μH21-H2Hn+11-Hn+1H2+kH2fV2-φH2Hn+1-φD2In-m2+1Dn+1I2-φI2Hn-m1+1fVn-m1In+1H2fV2-φV2Dn+1Vn+1D2+fVn+1fV2-Vn+1V2+ln⁡fV2Vn+1fVn+1V2=μH21-H2Hn+11-Hn+1H2+kH2fV2-φH2Hn+1-φD2In-m2+1Dn+1I2-φI2Hn-m1+1fVn-m1In+1H1fV2-φV2Dn+1Vn+1D2-φfV2Vn+1V2fVn+1+1-fV2fVn+1fVn+1fV2-Vn+1V2.Similar to the proof of Theorem 10, if R1>1, then we have Wn+1(3)-Wn(3)≤0, for all n∈N. That is, Wn(3) is monotone decreasing sequence. Furthermore, since Wn(3)>0, there is a limit limn→∞Wn(3)≥0. Hence, limn→∞(Wn+1(3)-Wn(3))=0. Combined with model (6), it can be shown that limn→∞Hn=H2, limn→∞In=I2, limn→∞Dn=D2, limn→∞Vn=V2, limn→∞Zn=Z2, which implies that E2 is globally asymptotically stable. This completes the proof.


## 4. Conclusions

In this paper, we proposed and investigated a delayed virus infection model with immune response and general nonlinear incidence. To come up with the efficient numerical method for the proposed delayed model, we then consider the discretization of the original continuous model by utilizing NSFD scheme. The advantage of the NSFD scheme is that the global properties of the solutions for the corresponding continuous model can be preserved. A crucial observation regarding the advantage of the NSFD scheme is that the discrete model has equilibria which are exactly the same as those of the original continuous model and the conditions for their stability are identical in case of both the continuous and discrete models. Specifically, ifR0≤1, then the infection-free equilibrium E0 is globally asymptotically stable; if R1≤1<R0, then the infection equilibrium without immunity E1 is globally asymptotically stable; if R1>1, then the infection equilibrium with immunity E2 is globally asymptotically stable. The results imply that the NSFD scheme can efficiently preserve the global properties of solutions for original continuous model [8]. Applying this method to the other types of delayed epidemic models is our future work.



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*Source: 3141736-2017-11-07.xml*