# Global Properties of a General Latent Pathogen Dynamics Model with Delayed Pathogenic and Cellular Infections

Authors: A. M. Elaiw; A. A. Almatrafi; A. D. Hobiny; K. Hattaf
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

This paper studies the global dynamics of a general pathogenic infection model with two ways of infections. The effect of antibody immune response is analyzed. We incorporate three discrete time delays and both latently infected cells and actively infected cells. The infection rate and production and clearance/death rates of the cells and pathogens are given by general functions. We determine two threshold parameters to investigate the global stability of three equilibria. We use Lyapunov method to establish the global stability. We support our theoretical results by numerical simulations.


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## Body

## 1. Introduction

The study of mathematical modeling and analysis of pathogen infection is helpful for understanding within-host pathogen dynamics; therefore, it has attracted the attention of many researchers during the last decades [1–19]. In [1], Nowak and Bangham proposed a simple model for the interaction between a replicating virus and host cells. In the same work, they proposed a second model that improves the first model by including the cytotoxic T lymphocytes (CTL) immune response against infected cells. Several extensions of [1] without immunity are given in [2–16] and with CTL immune response in [18, 19]. Antibodies which are produced from the B cells play a prominent role in pathogen dynamics. The function of antibodies is to attack and kill pathogens. Based on the first simple model introduced in [1] and the fact that antibody immune response is more effective than CTL in some infections [20], Murase et al. [21] proposed a mathematical model that describes the interactions between the target cells, the pathogens, and the antibody immune response. This model is given as follows:(1)S˙t=ϰ-dSt-b1StPt,(2)A˙t=b1StPt-λAt,(3)P˙t=nλAt-cPt-qPtZt,(4)Z˙t=rPtZt-mZt,where S(t), A(t), P(t), and Z(t) are the concentrations of susceptible cells, actively infected cells, pathogens and antibodies, respectively. In the literature, several modifications have been done on model (1)-(4) (see, e.g., [22–29]).In model (1)-(4), the infection rate is given by bilinear incidence which may be not reasonable to describe the dynamics of the pathogen during the several stages of infection [4]. Pathogen dynamics models with antibodies and general incidence rate have been studied in several works (see, e.g., [24–26]), but with only pathogen-to-cell transmission. In [30], the authors considered only cell-to-cell transmission in order to model the interaction of the healthy and infected cells. Both ways of transmissions have been considered in [31–37], but without humoral immunity. In 2017, Lin et al. [38] presented a mathematical model with humoral immunity and both ways of transmissions that are modeled by two bilinear functions.Although antiviral drug therapies can significantly limit the level of viruses in the blood, there is still a low viral load due to ongoing latently infected cells reactivation. These cells play the role of viral reservoirs and are considered a major hurdle to virus clearance in patients under treatment. Variant models have been developed to study the dynamics of pathogens in the presence of latent reservoirs (see, e.g., [24, 39–43]). Recently, Elaiw et al. [44] have studied the global stability of a general pathogen dynamics model with delay and with both cellular and pathogenic infections by assuming that all the infected cells are active. In [45], a class of latently infected cells has been added in pathogen dynamics with two routes of infections but with specific forms of the incidence rate and production and clearance/death rates of the pathogens and cells.The main objective of this study is to develop a general pathogenic infection model with two ways of infections, antibody immune response, three discrete time delays, and both latently infected cells and actively infected cells. This model is described by the following nonlinear system:(5)S˙t=FSt-K1Pt+K2AtgSt,(6)L˙t=ϱe-ε1ω1K1Pt-ω1+K2At-ω1gSt-ω1-α+λLξLt,(7)A˙t=1-ϱe-ε2ω2K1Pt-ω2+K2At-ω2gSt-ω2-λψAt+αξLt,(8)P˙t=nλe-ε3ω3ψAt-ω3-cφ1Pt-qφ1Ptφ2Zt,(9)Z˙t=rφ1Ptφ2Zt-mφ2Zt,where the fractions ϱ and (1-ϱ) with 0<ϱ<1 are the proportions of infection that lead to latency and actively, respectively. The functions F, g, K1, K2, ξ, ψ, φ1, and φ2 are continuously differentiable and satisfy the following hypotheses:(H1) (i) there existsS0>0 such that FS0=0,FS>0 for S∈[0,S0),(ii)F′S<0 for all S>0,(iii) there existsz>0 and z¯>0 such that FS≤z-z¯S for all S≥0.(H2) (i)Ki(l),ξ(l),ψ(l),φi(l),g(l)>0 for all l>0 and Ki(0)=ξ(0)=ψ(0)=φi(0)=g(0)=0, i=1,2,(ii)Ki′(l)>0,ξ′(l)>0,ψ′(l)>0 and φi′(l)>0 for all l≥0, i=1,2, g′(l)>0 for all l>0,(iii) there arec1,c2,c3, and c4>0 such that ξ(l)≥c1l, ψ(l)≥c2l, φ1(l)≥c3l, and φ2(l)≥c4l for all l≥0.(H3)K1P/φ1(P)′≤0 and K2A/ψ(A)′≤0 for all P,A>0.Let the initial conditions for system (5)-(9) be given as(10)Sη=χ1η,Lη=χ2η,Aη=χ3η,Pη=χ4η,Zη=χ5η,(11)χjη≥0,η∈-κ,0,(12)χj∈C-κ,0,R≥0,j=1,…,5,where κ=max⁡ω1,ω2,ω3 and C is the Banach space of continuous functions mapping the interval [-κ,0] into R≥0 with norm χj=sup-κ≤η≤0⁡χj(η). This shows the uniqueness of the solution for t>0.


### 1.1. Properties of Solution

Lemma 1.
Let (H1), (H2), and (H3) be valid. Then, the solutions of system (5)-(9) with initial conditions (11) are nonnegative and ultimately bounded.Proof.
System (5)-(9) can be written as dXt/dt=YXt, where(13)Xt=StLtAtPtZt,Y=Y1Y2Y3Y4Y5,and(14)YXt=Y1XtY2XtY3XtY4XtY5Xt=FSt-Δtϱe-ε1ω1Δt-ω1-α+λLξLt1-ϱe-ε2ω2Δt-ω2-λψAt+αξLtnλe-ε3ω3ψAt-ω3-cφ1Pt-qφ1Ptφ2Ztrφ1Ptφ2Zt-mφ2Zt,where(15)Δt=K1Pt+K2AtgSt.It follows that(16)YiXtXit=0,Xt∈R>05,≥0,i=1,…,5.Any solution of system (5)-(9) with the initial conditions (11) satisfies Xt∈R≥05 for all t≥0 [46]. Hence, R≥05 is positively invariant for system (5)-(9).
From (5), we obtain S˙t≤FS(t)≤z-z¯St. It follows that limt→∞⁡sup⁡S(t)≤z/z¯. Let F1G(t)=ϱe-ε1ω1S(t-ω1)+(1-ϱ)e-ε2ω2S(t-ω2)+L(t)+A(t). Then (17)F1G˙t=ϱe-ε1ω1FSt-ω1-Δt-ω1+1-ϱe-ε2ω2FSt-ω2-Δt-ω2+ϱe-ε1ω1Δt-ω1-α+λLξLt+1-ϱe-ε2ω2Δt-ω2-λψAt+αξLt=ϱe-ε1ω1FSt-ω1+1-ϱe-ε2ω2FSt-ω2-λLξLt-λψAt≤ϱe-ε1ω1z-z¯St-ω1+1-ϱe-ε2ω2z-z¯St-ω2-λLc1Lt-λc2At≤z-σ1GF1Gt,where σ1G=min⁡z¯,λLc1,λc2. It follows that limt→∞⁡sup⁡F1G(t)≤ς1G, where ς1G=z/σ1G, and then limt→∞⁡sup⁡L(t)≤ς1G and limt→∞⁡sup⁡A(t)≤ς1G. Moreover, let F2G(t)=P(t)+(q/r)Z(t). Then(18)F2G˙t=nλe-ε3ω3ψAt-ω3-cφ1P-qmrφ2Z≤nλe-ε3ω3ψς1G-cc3P-qmrc4Z≤nλψς1G-σ2GF2Gt,where σ2G=min⁡cc3,mc4. Then, limt→∞⁡sup⁡F2G(t)≤ς2G, where ς2G=nλψς1G/σ2G. This yields limt→∞⁡sup⁡P(t)≤ς2G and limt→∞⁡sup⁡Z(t)≤ς3G, where ς3G=(r/q)ς2G. Then, S(t), L(t), A(t), P(t), and Z(t) are ultimately bounded.Define the feasible region(19)ΓG=S,L,A,P,Z∈C:S≤ς1G,L≤ς1G,A≤ς1G,P≤ς2G,Z≤ς3G.Clearly, ΓG contains omega limit sets of system (5)-(9) and is also positively invariant for the system.


### 1.2. Equilibria

Lemma 2.
Let (H1)-(H3) be satisfied; then there exist two bifurcation numbersR0G and R1G>0 such that
(i) there exists only one equilibriumΘ0 if R0G≤1 and R1G≤1,
(ii) there exist only two equilibriaΘ0 and Θ1 if R1G≤1<R0G, and
(iii) there exist three equilibriaΘ0,Θ1, and Θ2 if R0G>1 and R1G>1.Proof.
LetΘS,L,A,P,Z be any equilibrium satisfying(20)0=FS-K1P+K2AgS,(21)0=ϱe-ε1ω1K1P+K2AgS-α+λLξL,(22)0=1-ϱe-ε2ω2K1P+K2AgS-λψA+αξL,(23)0=nλe-ε3ω3ψA-cφ1P-qφ1Pφ2Z,(24)0=rφ1Pφ2Z-mφ2Z.From (24), we have either φ2Z=0 or φ2Z≠0. First, we consider φ2Z=0; then using (H2), we obtain Z=0. From (20)-(23), we obtain(25)FS=K1P+K2AgS=α+λLξLϱe-ε1ω1=λe-ε3ω3γψA=cγnφ1P,where(26)γ=αϱα+λLe-ε1ω1-ε3ω3+1-ϱe-ε2ω2-ε3ω3.Hypothesis (H2) implies that the following functions:(27)L=ξ-1ϱe-ε1ω1α+λLFS,A=ψ-1γλe-ε3ω3FS,P=φ1-1γncFS,are strictly increasing and continuous.
Substituting (27) into (20), we get(28)K1φ1-1γncFS+K2ψ-1γλe-ε3ω3FSgS-FS=0.Let(29)H1GS=K1φ1-1γncFS+K2ψ-1γλe-ε3ω3FSgS-FS=0.According to (H1) and (H2), we have H1G0=-F0<0 and H1GS0=0.
SubstitutingS=S0 in (27), we get L=0, A=0, and P=0. This yields pathogen-free equilibrium Θ0=S0,0,0,0,0. Now from (29), we get(30)H1G′S=K1P+K2Ag′S+K1′PdPdS+K2′AdAdSgS-F′S.Moreover, from (25), we have(31)dAdS=γλe-ε3ω3F′Sψ′A,dPdS=γncF′Sφ1′P.Therefore, (30) becomes(32)H1G′S0=K10+K20g′S0+K1′0γnF′S0cφ1′0+K2′0γF′S0λe-ε3ω3ψ′0gS0-F′S0.Hypothesis (H2) implies that(33)H1G′S0=F′S0γnK1′0gS0cφ1′0+γK2′0gS0λe-ε3ω3ψ′0-1.From (H1), we have F′S0<0. Since H1G0<0 and H1GS0=0, hence, if(34)γnK1′0gS0cφ1′0+γK2′0gS0λe-ε3ω3ψ′0>1,then H1G′S0<0 and there exists S1∈0,S0 such that H1GS1=0. If inequality (34) is satisfied, then from (27) and (H1)-(H2), we have(35)L1=ξ-1ϱe-ε1ω1α+λLFS1>0,A1=ψ-1γλe-ε3ω3FS1>0,P1=φ1-1γncFS1>0.It follows that an infected equilibrium without antibodies Θ1=S1,L1,A1,P1,0 exists when condition (34) is satisfied. Therefore, we can define the basic reproduction number for system (5)-(9) as(36)R0G=γgS0nK1′0cφ1′0+K2′0λe-ε3ω3ψ′0.Let R0G=R01G+R02G, where(37)R01G=nγgS0K1′0cφ1′0,R02G=γgS0K2′0λe-ε3ω3ψ′0The other possibility of (24) is φ2Z≠0 and(38)P2=φ1-1mr>0.Let P=P2 in (20)-(22) and define function H2GS as(39)H2GS=K1P2+K2ψ-1γλe-ε3ω3FSgS-FS=0.Based on (H1) and (H2), we have H2G0=-F0<0 and H2GS0=K1P2gS0>0. Hence, there exists S2∈(0,S0) such that H2GS2=0. Thus, L2=ξ-1(ϱe-ε1ω1/(α+λL))FS2>0 and A2=ψ-1(γ/λe-ε3ω3)FS2>0. Now, from (23), we have(40)Z2=φ2-1cqrnλe-ε3ω3cmψA2-1.From (H2), we have if (rnλe-ε3ω3/cm)ψA2>1, then Z2>0. Now we can define(41)R1G=rnλe-ε3ω3cmψA2=rnγcmFS2,which represents the antibody immune response activation number. We have Z2=φ2-1(c/q)R1G-1; then if R1G>1, there exists an infected equilibrium with antibodies Θ2=S2,L2,A2,P2,Z2.


### 1.3. Global Properties

It should be noted that finding an appropriate Lyapunov functional for higher-order nonlinear delay differential equations is a more difficult task. For the construction of Lyapunov functional of fifth-order nonlinear differential equations with multiple deviating arguments, we refer the reader to the work [47]. Here, we study the global stability of equilibria of system (5)-(9) by means of direct Lyapunov method. We first define a function G(η)=η-1-ln⁡η and use the notation (S,L,A,P,Z) = (S(t),L(t),A(t),P(t),Z(t)).Theorem 3.
For system (5)-(9), suppose that R0G≤1 and (H1)-(H3) be held true; then Θ0 is globally asymptotically stable (GAS).Proof.
DefineU0G(S,L,A,P,Z) as(42)U0G=γS-S0-∫S0SgS0gηdη+αα+λLe-ε3ω3L+e-ε3ω3A+1-R02GnP+q1-R02GrnZ+e-ε1ω1-ε3ω3αϱα+λL∫0ω1K1Pt-η+K2At-ηgSt-ηdη+e-ε2ω2-ε3ω31-ϱ∫0ω2K1Pt-η+K2At-ηgSt-ηdη+e-ε3ω31-R02Gλ∫0ω3ψAt-ηdη,where γ is defined by (26). It is seen that U0G(S,L,A,P,Z)>0 for all S,L,A,P,Z>0, while at the pathogen-free equilibrium Θ0, we get U0G(S0,0,0,0,0)=0. Calculating dU0G/dt along system (5)-(9), we obtain (43)dU0Gdt=γ1-gS0gSFS-K1P+K2AgS+e-ε3ω3αα+λLϱe-ε1ω1K1Pt-ω1+K2At-ω1gSt-ω1-α+λLξL+e-ε3ω31-ϱe-ε2ω2K1Pt-ω2+K2At-ω2gSt-ω2-λψA+αξL+1-R02Gnnλe-ε3ω3ψAt-ω3-cφ1P-qφ1Pφ2Z+q1-R02Grnrφ1Pφ2Z-mφ2Z+e-ε1ω1-ε3ω3αϱα+λLK1P+K2AgS-K1Pt-ω1+K2At-ω1gSt-ω1+e-ε2ω2-ε3ω31-ϱK1P+K2AgS-K1Pt-ω2+K2At-ω2gSt-ω2+λe-ε3ω31-R02GψA-ψAt-ω3.Collecting terms of (43), we get (44)dU0Gdt=γ1-gS0gSFS-FS0+γK1P+K2AgS0-cn1-R02Gφ1P-qmrn1-R02Gφ2Z-λe-ε3ω3R02GψA.From (H3), we have (45)K1Pφ1P≤limP→0+⁡K1Pφ1P=K1′0φ1′0⇒K1P≤K1′0φ1Pφ1′0,K2AψA≤limA→0+⁡K2AψA=K2′0ψ′0⇒K2A≤K2′0ψAψ′0.It follows that (46)dU0Gdt≤γ1-gS0gSFS-FS0+γK1′0gS0φ1′0φ1P+γK2′0gS0ψ′0ψA-cn1-R02Gφ1P-qmrn1-R02Gφ2Z-λe-ε3ω3R02GψA.We have(47)γK1′0gS0φ1′0-cn1-R02G=cnR0G-1,γK2′0gS0ψ′0-λe-ε3ω3R02G=0.Therefore, we obtain(48)dU0Gdt≤γ1-gS0gSFS-FS0+cnR0G-1φ1P+qmrnR02G-1φ2Z.From (H1) and (H2), we have(49)γ1-gS0gSFS-FS0≤0.Therefore, if R0G≤1, then dU0G/dt≤0 for all S,P,Z>0. Moreover, dU0G/dt=0 when Z(t)=0, P(t)=0, and S(t)=S0. The solutions of system (5)-(9) tend to D0G′ the largest invariant subset of D0G=S,L,A,P,Z:dU0G/dt=0 [48]. For each solution in D0G′ satisfies P(t)=0 and from (8), we get(50)P˙t=0=nλe-ε3ω3ψAt-ω3.Then, A(t)=0. Similarly, from (7), we get L(t)=0. It follows that D0G′=Θ0. From LaSalle’s invariance principle, Θ0 is GAS when R0G≤1.Remark 4.
From (H1)-(H3), we have(51)K1Pφ1P-K1P1φ1P1K1P-K1P1≤0,K2AψA-K2A1ψA1K2A-K2A1≤0.Thus,(52)K1PK1P1-φ1Pφ1P11-K1P1K1P≤0,K2AK2A1-ψAψA11-K2A1K2A≤0.Obviously, the equilibriumΘ1 does not exist if R0G<1, and Θ1=Θ0 if R0G=1. When R0G>1, we have the following theorem.Theorem 5.
For system (5)-(9), assume that R1G≤1<R0G and let (H1)-(H3) be held true; then, Θ1 is GAS.Proof.
LetU1G(S,L,A,P,Z) be given as follows:(53)U1G=γS-S1-∫S1SgS1gηdη+αe-ε3ω3α+λLL-L1-∫L1LξL1ξηdη+e-ε3ω3A-A1-∫A1AψA1ψηdη+γK1P1gS1nλe-ε3ω3ψA1P-P1-∫P1Pφ1P1φ1ηdη+γqK1P1gS1rnλe-ε3ω3ψA1Z+e-ε1ω1-ε3ω3αϱα+λLK1P1gS1∫0ω1GK1Pt-ηgSt-ηK1P1gS1dη+e-ε2ω2-ε3ω31-ϱK1P1gS1∫0ω2GK1Pt-ηgSt-ηK1P1gS1dη+e-ε1ω1-ε3ω3αϱα+λLK2A1gS1∫0ω1GK2At-ηgSt-ηK2A1gS1dη+e-ε2ω2-ε3ω31-ϱK2A1gS1∫0ω2GK2At-ηgSt-ηK2A1gS1dη+γK1P1gS1∫0ω3GψAt-ηψA1dη.We have U1G(S,L,A,P,Z)>0 for all S,L,A,P,Z>0 and U1G(S1,L1,A1,P1,0)=0. Calculating dU1G/dt, we obtain(54)dU1Gdt=γ1-gS1gSFS-K1P+K2AgS+e-ε3ω3αα+λL1-ξL1ξLϱe-ε1ω1K1Pt-ω1+K2At-ω1gSt-ω1-α+λLξL+e-ε3ω31-ψA1ψA1-ϱe-ε2ω2K1Pt-ω2+K2At-ω2gt-ω2-λψA+αξL+γK1P1gS1nλe-ε3ω3ψA11-φ1P1φ1Pnλe-ε3ω3ψAt-ω3-cφ1P-aφ1Pφ2Z+γqK1P1gS1rnλe-ε3ω3ψA1rφ1Pφ2Z-mφ2Z+e-ε1ω1-ε3ω3αϱα+λLK1P1gS1K1PgSK1P1gS1-K1Pt-ω1gSt-ω1K1P1gS1+ln⁡K1Pt-ω1gSt-ω1K1PgS+e-ε2ω2-ε3ω31-ϱK1P1gS1K1PgSK1P1gS1-K1Pt-ω2gSt-ω2K1P1gS1+ln⁡K1Pt-ω2gSt-ω2K1PgS+e-ε1ω1-ε3ω3αϱα+λLK2A1gS1K2AgSK2A1gS1-K2At-ω1gSt-ω1K2A1gS1+ln⁡K2At-ω1gSt-ω1K2AgS+e-ε2ω2-ε3ω31-ϱK2A1gS1K2AgSK2A1gS1-K2At-ω2gSt-ω2K2A1gS1+lnK2At-ω2gSt-ω2K2AgS+γK1P1gS1ψAψA1-ψAt-ω3ψA1+ln⁡ψAt-ω3ψA.Simplifying (54), we get (55)dU1Gdt=γ1-gS1gSFS+γK1P+K2AgS1-e-ε1ω1-ε3ω3αϱα+λLK1Pt-ω1gSt-ω1ξL1ξL+K2At-ω1gSt-ω1ξL1ξL+αe-ε3ω3ξL1-λe-ε3ω3ψA-e-ε2ω2-ε3ω31-ϱK1Pt-ω2gSt-ω2ψA1ψA+K2At-ω2gSt-ω2ψA1ψA+λe-ε3ω3ψA1-αe-ε3ω3ξLψA1ψA-γK1P1gS1cφ1Pnλe-ε3ω3ψA1+ψAt-ω3φ1P1ψA1φ1P-cφ1P1nλe-ε3ω3ψA1-qφ1P1φ2Znλe-ε3ω3ψA1+qmφ2Zrnλe-ε3ω3ψA1+e-ε1ω1-ε3ω3αϱα+λLK1P1gS1ln⁡K1Pt-ω1gSt-ω1K1PgS+e-ε2ω2-ε3ω31-ϱK1P1gS1ln⁡K1Pt-ω2gSt-ω2K1PgS+e-ε1ω1-ε3ω3αϱα+λLK2A1gS1ln⁡K2At-ω1gSt-ω1K2AgS+e-ε2ω2-ε3ω31-ϱK2A1gS1ln⁡K2At-ω2gSt-ω2K2AgS+γK1P1gS1ln⁡ψAt-ω3ψA+γK1P1gS1ψAψA1.The components of Θ1 satisfy (56)FS1=K1P1+K2A1gS1,e-ε1ω1-ε3ω3αϱα+λLK1P1+K2A1gS1=αe-ε3ω3ξL1,γK1P1+K2A1gS1=λe-ε3ω3ψA1,φ1P1=nλe-ε3ω3ψA1c.We get (57)dU1Gdt=γ1-gS1gSFS-FS1+γK1P1+K2A1gS1-γK1P1+K2A1gS1gS1gS+γK1P+K2AgS1-e-ε1ω1-ε3ω3αϱα+λLK1Pt-ω1gSt-ω1ξL1ξL+K2At-ω1gSt-ω1ξL1ξL+e-ε1ω1-ε3ω3αϱα+λLK1P1+K2A1gS1-γK2A1gS1ψAψA1-e-ε2ω2-ε3ω31-ϱK1Pt-ω2gSt-ω2ψA1ψA+K2At-ω2gSt-ω2ψA1ψA+γK1P1+K2A1gS1-e-ε1ω1-ε3ω3αϱα+λLK1P1+K2A1gS1ξLψA1ξL1ψA-γK1P1gS1φ1Pφ1P1+ψAt-ω3φ1P1ψA1φ1P-1+γqK1P1gS1cφ1P1φ1P1-φ1P2φ2Z+e-ε1ω1-ε3ω3αϱα+λLK1P1gS1ln⁡K1Pt-ω1gSt-ω1K1PgS+e-ε2ω2-ε3ω31-ϱK1P1gS1ln⁡K1Pt-ω2gSt-ω2K1PgS+e-ε1ω1-ε3ω3αϱα+λLK2A1gS1ln⁡K2At-ω1gSt-ω1K2AgS+e-ε2ω2-ε3ω31-ϱK2A1gS1ln⁡K2At-ω2gSt-ω2K2AgS+γK1P1gS1ln⁡ψAt-ω3ψA.Using the equalities in case of i=1,(58)ln⁡K1Pt-ω1gSt-ω1K1PgS=ln⁡K1Pt-ω1gSt-ω1ξLiK1PigSiξL+ln⁡ξLψAiξLiψA+ln⁡ψAK1PiψAiK1P+ln⁡gSigS,ln⁡K1Pt-ω2gSt-ω2K1PgS=ln⁡K1Pt-ω2gSt-ω2ψAiK1PigSiψA+ln⁡ψAK1PiψAiK1P+ln⁡gSigS,ln⁡K2At-ω1gSt-ω1K2AgS=ln⁡K2At-ω1gSt-ω1ξLiK2AigSiξL+ln⁡ξLψAiξLiψA+ln⁡ψAK2AiψAiK2A+ln⁡gSigS,ln⁡K2At-ω2gSt-ω2K2AgS=ln⁡K2At-ω2gSt-ω2ψAiK2AigSiψA+ln⁡ψAK2AiψAiK2A+ln⁡gSigS,ln⁡ψAt-ω3ψA=ln⁡ψAt-ω3φ1PiψAiφ1P+ln⁡ψAiφ1PψAφ1Pi,Therefore, we obtain (59)dU1Gdt=γ1-gS1gSFS-FS1+γK1P1gS1K1P1φ1PK1Pφ1P1+K1PK1P1-1-φ1Pφ1P1+γK2A1gS1ψAK2A1ψA1K2A+K2AK2A1-1-ψAψA1-γK1P1+K2A1gS1gS1gS-1-ln⁡gS1gS-e-ε1ω1-ε3ω3αϱα+λLK1P1gS1K1Pt-ω1gSt-ω1ξL1K1P1gS1ξL-1-ln⁡K1Pt-ω1gSt-ω1ξL1K1P1gS1ξL-e-ε1ω1-ε3ω3αϱα+λLK2A1gS1K2At-ω1gSt-ω1ξL1K2A1gS1ξL-1-ln⁡K2At-ω1gSt-ω1ξL1K2A1gS1ξL-e-ε2ω2-ε3ω31-ϱK1P1gS1K1Pt-ω2gSt-ω2ψA1K1P1gS1ψA-1-ln⁡K1Pt-ω2gSt-ω2ψA1K1P1gS1ψA-e-ε2ω2-ε3ω31-ϱK2A1gS1K2At-ω2gSt-ω2ψA1K2A1gS1ψA-1-lnK2At-ω2gSt-ω2ψA1K2A1gS1ψA-e-ε1ω1-ε3ω3αϱα+λLK1P1+K2A1gS1ξLψA1ξL1ψA-1-ln⁡ξLψA1ξL1ψA-γK1P1gS1ψAt-ω3φ1P1ψA1φ1P-1-ln⁡ψAt-ω3φ1P1ψA1φ1P-γK1P1gS1K1P1φ1PK1Pφ1P1-1-ln⁡K1P1φ1PK1Pφ1P1-γK2A1gS1ψAK2A1ψA1K2A-1-ln⁡ψAK2A1ψA1K2A+γqK1P1gS1cφ1P1φ1P1-φ1P2φ2Z.dU1Gdt=γ1-gS1gSFS-FS1+γK1P1gS1K1PK1P1-φ1Pφ1P11-K1P1K1P+γK2A1gS1K2AK2A1-ψAψA11-K2A1K2A-γK1P1+K2A1gS1GgS1gS-e-ε1ω1-ε3ω3αϱα+λLK1P1gS1GK1Pt-ω1gSt-ω1ξL1K1P1gS1ξL-e-ε1ω1-ε3ω3αϱα+λLK2A1gS1GK2At-ω1gSt-ω1ξL1K2A1gS1ξL-e-ε2ω2-ε3ω31-ϱK1P1gS1GK1Pt-ω2gSt-ω2ψA1K1P1gS1ψA-e-ε2ω2-ε3ω31-ϱK2A1gS1GK2At-ω2gSt-ω2ψA1K2A1gS1ψA-e-ε1ω1-ε3ω3αϱα+λLK1P1+K2A1gS1GξLψA1ξL1ψA-γK1P1gS1GψAt-ω3φ1P1ψA1φ1P-γK1P1gS1GK1P1φ1PK1Pφ1P1-γK2A1gS1GψAK2A1ψA1K2A+γqK1P1gS1cφ1P1φ1P1-φ1P2φ2Z.If R1G≤1, then Θ2 does not exist since Z2=φ2-1(c/q)R1G-1≤0. This means Zt=rφ1P-mφ2Z=rφ1P-φ1P2φ2Z≤0, for all Z>0. Thus, φ1P≤φ1P2 for all P≥0. It follows that φ1P1≤φ1P2. Then according to (H1)-(H3), we have dU1G/dt≤0. The solutions of system (5)-(9) tend to D1G′ the largest invariant subset of D1G=S,L,A,P,Z:dU1G/dt=0. It is easy to show that D1G′=Θ1. Applying LaSalle’s invariance principle, we get that if R1G≤1 and R0G>1, then Θ1 is GAS.On the other hand, it is not hard to verify that the equilibriumΘ2 does not exist if R1G<1, and Θ2=Θ1 if R1G=1. When R1G>1, we have the following result.Theorem 6.
For system (5)-(9), suppose that R1G>1 and let (H1)-(H3) be held true; then, Θ2 is GAS.Proof.
ConsiderU2G(S,L,A,P,Z)(60)U2G=γS-S2-∫S2SgS2gηdη+αe-ε3ω3α+λLL-L2-∫L2LξL2ξηdη+e-ε3ω3A-A2-∫A2AψA2ψηdη+γK1P2gS2nλe-ε3ω3ψA2P-P2-∫P2Pφ1P2φ1ηdη+γqK1P2gS2rnλe-ε3ω3ψA2Z-Z2-∫Z2Zφ2Z2φ2ηdη+e-ε1ω1-ε3ω3αϱα+λLK1P2gS2∫0ω1GK1Pt-ηgSt-ηK1P2gS2dη+e-ε2ω2-ε3ω31-ϱK1P2gS2∫0ω2GK1Pt-ηgSt-ηK1P2gS2dη+e-ε1ω1-ε3ω3αϱα+λLK2A2gS2∫0ω1GK2At-ηgSt-ηK2A2gS2dη+e-ε2ω2-ε3ω31-ϱK2A2gS2∫0ω2GK2At-ηgSt-ηK2A2gS2dη+γK1P2gS2∫0ω3GψAt-ηψA2dη.We have U2G(S,L,A,P,Z)>0 for all S,L,A,P,Z>0, while U2G reaches its global minimum at Θ2. Calculate dU2G/dt as(61)dU2Gdt=γ1-gS2gSFS-K1P+K2AgS+e-ε3ω3αα+λL1-ξL2ξLϱe-ε1ω1K1Pt-ω1+K2At-ω1gSt-ω1-α+λLξL+e-ε3ω31-ψA2ψA1-ϱe-ε2ω2K1Pt-ω2+K2At-ω2gSt-ω2-λψA+αξL+γK1P2gS2nλe-ε3ω3ψA21-φ1P2φ1Pnλe-ε3ω3ψAt-ω3-cφ1P-qφ1Pφ2Z+γqK1P2gS2rnλe-ε3ω3ψA21-φ2Z2φ2Zrφ1Pφ2Z-mφ2Z+e-ε1ω1-ε3ω3αϱα+λLK1P2gS2K1PgSK1P2gS2-K1Pt-ω1gSt-ω1K1P2gS2+ln⁡K1Pt-ω1gSt-ω1K1PgS+e-ε2ω2-ε3ω31-ϱK1P2gS2K1PgSK1P2gS2-K1Pt-ω2gSt-ω2K1P2gS2+ln⁡K1Pt-ω2gSt-ω2K1PgS+e-ε1ω1-ε3ω3αϱα+λLK2A2gS2K2AgSK2A2gS2-K2At-ω1gSt-ω1K2A2gS2+ln⁡K2At-ω1gSt-ω1K2AgS+e-ε2ω2-ε3ω31-ϱK2A2gS2K2AgSK2A2gS2-K2At-ω2gSt-ω2K2A2gS2+ln⁡K2At-ω2gSt-ω2K2AgS+γK1P2gS2ψAψA2-ψAt-ω3ψA2+ln⁡ψAt-ω3ψA.Simplify (61) as (62)dU2Gdt=γ1-gS2gSFS+γK1P+K2AgS2-e-ε1ω1-ε3ω3αϱα+λLK1Pt-ω1gSt-ω1ξL2ξL+K2At-ω1gSt-ω1ξL2ξL+αe-ε3ω3ξL2-λe-ε3ω3ψA-e-ε2ω2-ε3ω31-ϱK1Pt-ω2gSt-ω2ψA2ψA+K2At-ω2gSt-ω2ψA2ψA+λe-ε3ω3ψA2-αe-ε3ω3ξLψA2ψA-γK1P2gS2cφ1Pnλe-ε3ω3ψA2+ψAt-ω3φ1P2ψA2φ1P-cφ1P2nλe-ε3ω3ψA2-qφ1P2φ2Znλe-ε3ω3ψA2+qmφ2Zrnλe-ε3ω3ψA2+qφ1Pφ2Z2nλe-ε3ω3ψA2-qmφ2Z2rnλe-ε3ω3ψA2+e-ε1ω1-ε3ω3αϱα+λLK1P2gS2ln⁡K1Pt-ω1gSt-ω1K1PgS+e-ε2ω2-ε3ω31-ϱK1P2gS2ln⁡K1Pt-ω2gSt-ω2K1PgS+e-ε1ω1-ε3ω3αϱα+λLK2A2gS2ln⁡K2At-ω1gSt-ω1K2AgS+e-ε2ω2-ε3ω31-ϱK2A2gS2ln⁡K2At-ω2gSt-ω2K2AgS+γK1P2gS2ln⁡ψAt-ω3ψA+γK1P2gS2ψAψA2.The components of Θ2 satisfy(63)FS2=K1P2+K2A2gS2,e-ε1ω1-ε3ω3αϱα+λLK1P2+K2A2gS2=αe-ε3ω3ξL2,γK1P2+K2A2gS2=λe-ε3ω3ψA2,nλe-ε3ω3ψA2=cφ1P2+qφ1P2φ2Z2,φ1P2=mr,we get(64)dU2Gdt=γ1-gS2gSFS-FS2+γK1P2+K2A2gS2-γK1P2+K2A2gS2gS2gS+γK1P+K2AgS2-e-ε1ω1-ε3ω3αϱα+λLK1Pt-ω1gSt-ω1ξL2ξL+K2At-ω1gSt-ω1ξL2ξL+e-ε1ω1-ε3ω3αϱα+λLK1P2+K2A2gS2-γK2A2gS2ψAψA2-e-ε2ω2-ε3ω31-ϱK1Pt-ω2gSt-ω2ψA2ψA+K2At-ω2gSt-ω2ψA2ψA+e-ε1ω1-ε3ω3αϱα+λLK1P2+K2A2gS2+e-ε2ω2-ε3ω31-ϱK1P2+K2A2gS2-e-ε1ω1-ε3ω3αϱα+λLK1P2+K2A2gS2ξLψA2ξL2ψA-γK1P2gS2φ1Pφ1P2+ψAt-ω3φ1P2ψA2φ1P-1+e-ε1ω1-ε3ω3αϱα+λLK1P2gS2ln⁡K1Pt-ω1gSt-ω1K1PgS+e-ε2ω2-ε3ω31-ϱK1P2gS2ln⁡K1Pt-ω2gSt-ω2K1PgS+e-ε1ω1-ε3ω3αϱα+λLK2A2gS2ln⁡K2At-ω1gSt-ω1K2AgS+e-ε2ω2-ε3ω31-ϱK2A2gS2ln⁡K2At-ω2gSt-ω2K2AgS+γK1P2gS2ln⁡ψAt-ω3ψA.Using equalities (58) when i=2, we obtain (65)dU2Gdt=γ1-gS2gSFS-FS2+γK1P2gS2K1P2φ1PK1Pφ1P2+K1PK1P2-1-φ1Pφ1P2+γK2A2gS2K2A2ψAK2AψA2+K2AK2A2-1-ψAψA2-γK1P2+K2A2gS2gS2gS-1-ln⁡gS2gS-e-ε1ω1-ε3ω3αϱα+λLK1P2gS2K1Pt-ω1gSt-ω1ξL2K1P2gS2ξL-1-ln⁡K1Pt-ω1gSt-ω1ξL2K1P2gS2ξL-e-ε1ω1-ε3ω3αϱα+λLK2A2gS2K2At-ω1gSt-ω1ξL2K2A2gS2ξL-1-ln⁡K2At-ω1gSt-ω1ξL2K2A2gS2ξL-e-ε2ω2-ε3ω31-ϱK1P2gS2K1Pt-ω2gSt-ω2ψA2K1P2gS2ψA-1-ln⁡K1Pt-ω2gSt-ω2ψA2K1P2gS2ψA-e-ε2ω2-ε3ω31-ϱK2A2gS2K2At-ω2gSt-ω2ψA2K2A2gS2ψA-1-ln⁡K2At-ω2gSt-ω2ψA2K2A2gS2ψA-e-ε1ω1-ε3ω3αϱα+λLK1P2+K2A2gS2ξLψA2ξL2ψA-1-ln⁡ξLψA2ξL2ψA-γK1P2gS2ψAt-ω3φ1P2ψA2φ1P-1-ln⁡ψAt-ω3φ1P2ψA2φ1P-γK1P2gS2K1P2φ1PK1Pφ1P2-1-ln⁡K1P2φ1PK1Pφ1P2-γK2A2gS2K2A2ψAK2AψA2-1-ln⁡K2A2ψAK2AψA2.dU2Gdt=γ1-gS2gSFS-FS2+γK1P2gS2K1PK1P2-φ1Pφ1P21-K1P2K1P+γK2A2gS2K2AK2A2-ψAψA21-K2A2K2A-γK1P2+K2A2gS2GgS2gS-e-ε1ω1-ε3ω3αϱα+λLK1P2gS2GK1Pt-ω1gSt-ω1ξL2K1P2gS2ξL-e-ε1ω1-ε3ω3αϱα+λLK2A2gS2GK2At-ω1gSt-ω1ξL2K2A2gS2ξL-e-ε2ω2-ε3ω31-ϱK1P2gS2GK1Pt-ω2gSt-ω2ψA2K1P2gS2ψA-e-ε2ω2-ε3ω31-ϱK2A2gS2GK2At-ω2gSt-ω2ψA2K2A2gS2ψA-e-ε1ω1-ε3ω3αϱα+λLK1P2+K2A2gS2GξLψA2ξL2ψA-γK1P2gS2GψAt-ω3φ1P2ψA2φ1P-γK1P2gS2GK1P2φ1PK1Pφ1P2-γK2A2gS2GK2A2ψAK2AψA2.From (H1)-(H3), we have that dU2G/dt≤0 for all S,L,A,P and Z>0. One can show that dU2G/dt=0 at Θ2; then apply LaSalle’s invariance principle to show the global asymptotic stability of Θ2.


## 1.1. Properties of Solution

Lemma 1.
Let (H1), (H2), and (H3) be valid. Then, the solutions of system (5)-(9) with initial conditions (11) are nonnegative and ultimately bounded.Proof.
System (5)-(9) can be written as dXt/dt=YXt, where(13)Xt=StLtAtPtZt,Y=Y1Y2Y3Y4Y5,and(14)YXt=Y1XtY2XtY3XtY4XtY5Xt=FSt-Δtϱe-ε1ω1Δt-ω1-α+λLξLt1-ϱe-ε2ω2Δt-ω2-λψAt+αξLtnλe-ε3ω3ψAt-ω3-cφ1Pt-qφ1Ptφ2Ztrφ1Ptφ2Zt-mφ2Zt,where(15)Δt=K1Pt+K2AtgSt.It follows that(16)YiXtXit=0,Xt∈R>05,≥0,i=1,…,5.Any solution of system (5)-(9) with the initial conditions (11) satisfies Xt∈R≥05 for all t≥0 [46]. Hence, R≥05 is positively invariant for system (5)-(9).
From (5), we obtain S˙t≤FS(t)≤z-z¯St. It follows that limt→∞⁡sup⁡S(t)≤z/z¯. Let F1G(t)=ϱe-ε1ω1S(t-ω1)+(1-ϱ)e-ε2ω2S(t-ω2)+L(t)+A(t). Then (17)F1G˙t=ϱe-ε1ω1FSt-ω1-Δt-ω1+1-ϱe-ε2ω2FSt-ω2-Δt-ω2+ϱe-ε1ω1Δt-ω1-α+λLξLt+1-ϱe-ε2ω2Δt-ω2-λψAt+αξLt=ϱe-ε1ω1FSt-ω1+1-ϱe-ε2ω2FSt-ω2-λLξLt-λψAt≤ϱe-ε1ω1z-z¯St-ω1+1-ϱe-ε2ω2z-z¯St-ω2-λLc1Lt-λc2At≤z-σ1GF1Gt,where σ1G=min⁡z¯,λLc1,λc2. It follows that limt→∞⁡sup⁡F1G(t)≤ς1G, where ς1G=z/σ1G, and then limt→∞⁡sup⁡L(t)≤ς1G and limt→∞⁡sup⁡A(t)≤ς1G. Moreover, let F2G(t)=P(t)+(q/r)Z(t). Then(18)F2G˙t=nλe-ε3ω3ψAt-ω3-cφ1P-qmrφ2Z≤nλe-ε3ω3ψς1G-cc3P-qmrc4Z≤nλψς1G-σ2GF2Gt,where σ2G=min⁡cc3,mc4. Then, limt→∞⁡sup⁡F2G(t)≤ς2G, where ς2G=nλψς1G/σ2G. This yields limt→∞⁡sup⁡P(t)≤ς2G and limt→∞⁡sup⁡Z(t)≤ς3G, where ς3G=(r/q)ς2G. Then, S(t), L(t), A(t), P(t), and Z(t) are ultimately bounded.Define the feasible region(19)ΓG=S,L,A,P,Z∈C:S≤ς1G,L≤ς1G,A≤ς1G,P≤ς2G,Z≤ς3G.Clearly, ΓG contains omega limit sets of system (5)-(9) and is also positively invariant for the system.


## 1.2. Equilibria

Lemma 2.
Let (H1)-(H3) be satisfied; then there exist two bifurcation numbersR0G and R1G>0 such that
(i) there exists only one equilibriumΘ0 if R0G≤1 and R1G≤1,
(ii) there exist only two equilibriaΘ0 and Θ1 if R1G≤1<R0G, and
(iii) there exist three equilibriaΘ0,Θ1, and Θ2 if R0G>1 and R1G>1.Proof.
LetΘS,L,A,P,Z be any equilibrium satisfying(20)0=FS-K1P+K2AgS,(21)0=ϱe-ε1ω1K1P+K2AgS-α+λLξL,(22)0=1-ϱe-ε2ω2K1P+K2AgS-λψA+αξL,(23)0=nλe-ε3ω3ψA-cφ1P-qφ1Pφ2Z,(24)0=rφ1Pφ2Z-mφ2Z.From (24), we have either φ2Z=0 or φ2Z≠0. First, we consider φ2Z=0; then using (H2), we obtain Z=0. From (20)-(23), we obtain(25)FS=K1P+K2AgS=α+λLξLϱe-ε1ω1=λe-ε3ω3γψA=cγnφ1P,where(26)γ=αϱα+λLe-ε1ω1-ε3ω3+1-ϱe-ε2ω2-ε3ω3.Hypothesis (H2) implies that the following functions:(27)L=ξ-1ϱe-ε1ω1α+λLFS,A=ψ-1γλe-ε3ω3FS,P=φ1-1γncFS,are strictly increasing and continuous.
Substituting (27) into (20), we get(28)K1φ1-1γncFS+K2ψ-1γλe-ε3ω3FSgS-FS=0.Let(29)H1GS=K1φ1-1γncFS+K2ψ-1γλe-ε3ω3FSgS-FS=0.According to (H1) and (H2), we have H1G0=-F0<0 and H1GS0=0.
SubstitutingS=S0 in (27), we get L=0, A=0, and P=0. This yields pathogen-free equilibrium Θ0=S0,0,0,0,0. Now from (29), we get(30)H1G′S=K1P+K2Ag′S+K1′PdPdS+K2′AdAdSgS-F′S.Moreover, from (25), we have(31)dAdS=γλe-ε3ω3F′Sψ′A,dPdS=γncF′Sφ1′P.Therefore, (30) becomes(32)H1G′S0=K10+K20g′S0+K1′0γnF′S0cφ1′0+K2′0γF′S0λe-ε3ω3ψ′0gS0-F′S0.Hypothesis (H2) implies that(33)H1G′S0=F′S0γnK1′0gS0cφ1′0+γK2′0gS0λe-ε3ω3ψ′0-1.From (H1), we have F′S0<0. Since H1G0<0 and H1GS0=0, hence, if(34)γnK1′0gS0cφ1′0+γK2′0gS0λe-ε3ω3ψ′0>1,then H1G′S0<0 and there exists S1∈0,S0 such that H1GS1=0. If inequality (34) is satisfied, then from (27) and (H1)-(H2), we have(35)L1=ξ-1ϱe-ε1ω1α+λLFS1>0,A1=ψ-1γλe-ε3ω3FS1>0,P1=φ1-1γncFS1>0.It follows that an infected equilibrium without antibodies Θ1=S1,L1,A1,P1,0 exists when condition (34) is satisfied. Therefore, we can define the basic reproduction number for system (5)-(9) as(36)R0G=γgS0nK1′0cφ1′0+K2′0λe-ε3ω3ψ′0.Let R0G=R01G+R02G, where(37)R01G=nγgS0K1′0cφ1′0,R02G=γgS0K2′0λe-ε3ω3ψ′0The other possibility of (24) is φ2Z≠0 and(38)P2=φ1-1mr>0.Let P=P2 in (20)-(22) and define function H2GS as(39)H2GS=K1P2+K2ψ-1γλe-ε3ω3FSgS-FS=0.Based on (H1) and (H2), we have H2G0=-F0<0 and H2GS0=K1P2gS0>0. Hence, there exists S2∈(0,S0) such that H2GS2=0. Thus, L2=ξ-1(ϱe-ε1ω1/(α+λL))FS2>0 and A2=ψ-1(γ/λe-ε3ω3)FS2>0. Now, from (23), we have(40)Z2=φ2-1cqrnλe-ε3ω3cmψA2-1.From (H2), we have if (rnλe-ε3ω3/cm)ψA2>1, then Z2>0. Now we can define(41)R1G=rnλe-ε3ω3cmψA2=rnγcmFS2,which represents the antibody immune response activation number. We have Z2=φ2-1(c/q)R1G-1; then if R1G>1, there exists an infected equilibrium with antibodies Θ2=S2,L2,A2,P2,Z2.


## 1.3. Global Properties

It should be noted that finding an appropriate Lyapunov functional for higher-order nonlinear delay differential equations is a more difficult task. For the construction of Lyapunov functional of fifth-order nonlinear differential equations with multiple deviating arguments, we refer the reader to the work [47]. Here, we study the global stability of equilibria of system (5)-(9) by means of direct Lyapunov method. We first define a function G(η)=η-1-ln⁡η and use the notation (S,L,A,P,Z) = (S(t),L(t),A(t),P(t),Z(t)).Theorem 3.
For system (5)-(9), suppose that R0G≤1 and (H1)-(H3) be held true; then Θ0 is globally asymptotically stable (GAS).Proof.
DefineU0G(S,L,A,P,Z) as(42)U0G=γS-S0-∫S0SgS0gηdη+αα+λLe-ε3ω3L+e-ε3ω3A+1-R02GnP+q1-R02GrnZ+e-ε1ω1-ε3ω3αϱα+λL∫0ω1K1Pt-η+K2At-ηgSt-ηdη+e-ε2ω2-ε3ω31-ϱ∫0ω2K1Pt-η+K2At-ηgSt-ηdη+e-ε3ω31-R02Gλ∫0ω3ψAt-ηdη,where γ is defined by (26). It is seen that U0G(S,L,A,P,Z)>0 for all S,L,A,P,Z>0, while at the pathogen-free equilibrium Θ0, we get U0G(S0,0,0,0,0)=0. Calculating dU0G/dt along system (5)-(9), we obtain (43)dU0Gdt=γ1-gS0gSFS-K1P+K2AgS+e-ε3ω3αα+λLϱe-ε1ω1K1Pt-ω1+K2At-ω1gSt-ω1-α+λLξL+e-ε3ω31-ϱe-ε2ω2K1Pt-ω2+K2At-ω2gSt-ω2-λψA+αξL+1-R02Gnnλe-ε3ω3ψAt-ω3-cφ1P-qφ1Pφ2Z+q1-R02Grnrφ1Pφ2Z-mφ2Z+e-ε1ω1-ε3ω3αϱα+λLK1P+K2AgS-K1Pt-ω1+K2At-ω1gSt-ω1+e-ε2ω2-ε3ω31-ϱK1P+K2AgS-K1Pt-ω2+K2At-ω2gSt-ω2+λe-ε3ω31-R02GψA-ψAt-ω3.Collecting terms of (43), we get (44)dU0Gdt=γ1-gS0gSFS-FS0+γK1P+K2AgS0-cn1-R02Gφ1P-qmrn1-R02Gφ2Z-λe-ε3ω3R02GψA.From (H3), we have (45)K1Pφ1P≤limP→0+⁡K1Pφ1P=K1′0φ1′0⇒K1P≤K1′0φ1Pφ1′0,K2AψA≤limA→0+⁡K2AψA=K2′0ψ′0⇒K2A≤K2′0ψAψ′0.It follows that (46)dU0Gdt≤γ1-gS0gSFS-FS0+γK1′0gS0φ1′0φ1P+γK2′0gS0ψ′0ψA-cn1-R02Gφ1P-qmrn1-R02Gφ2Z-λe-ε3ω3R02GψA.We have(47)γK1′0gS0φ1′0-cn1-R02G=cnR0G-1,γK2′0gS0ψ′0-λe-ε3ω3R02G=0.Therefore, we obtain(48)dU0Gdt≤γ1-gS0gSFS-FS0+cnR0G-1φ1P+qmrnR02G-1φ2Z.From (H1) and (H2), we have(49)γ1-gS0gSFS-FS0≤0.Therefore, if R0G≤1, then dU0G/dt≤0 for all S,P,Z>0. Moreover, dU0G/dt=0 when Z(t)=0, P(t)=0, and S(t)=S0. The solutions of system (5)-(9) tend to D0G′ the largest invariant subset of D0G=S,L,A,P,Z:dU0G/dt=0 [48]. For each solution in D0G′ satisfies P(t)=0 and from (8), we get(50)P˙t=0=nλe-ε3ω3ψAt-ω3.Then, A(t)=0. Similarly, from (7), we get L(t)=0. It follows that D0G′=Θ0. From LaSalle’s invariance principle, Θ0 is GAS when R0G≤1.Remark 4.
From (H1)-(H3), we have(51)K1Pφ1P-K1P1φ1P1K1P-K1P1≤0,K2AψA-K2A1ψA1K2A-K2A1≤0.Thus,(52)K1PK1P1-φ1Pφ1P11-K1P1K1P≤0,K2AK2A1-ψAψA11-K2A1K2A≤0.Obviously, the equilibriumΘ1 does not exist if R0G<1, and Θ1=Θ0 if R0G=1. When R0G>1, we have the following theorem.Theorem 5.
For system (5)-(9), assume that R1G≤1<R0G and let (H1)-(H3) be held true; then, Θ1 is GAS.Proof.
LetU1G(S,L,A,P,Z) be given as follows:(53)U1G=γS-S1-∫S1SgS1gηdη+αe-ε3ω3α+λLL-L1-∫L1LξL1ξηdη+e-ε3ω3A-A1-∫A1AψA1ψηdη+γK1P1gS1nλe-ε3ω3ψA1P-P1-∫P1Pφ1P1φ1ηdη+γqK1P1gS1rnλe-ε3ω3ψA1Z+e-ε1ω1-ε3ω3αϱα+λLK1P1gS1∫0ω1GK1Pt-ηgSt-ηK1P1gS1dη+e-ε2ω2-ε3ω31-ϱK1P1gS1∫0ω2GK1Pt-ηgSt-ηK1P1gS1dη+e-ε1ω1-ε3ω3αϱα+λLK2A1gS1∫0ω1GK2At-ηgSt-ηK2A1gS1dη+e-ε2ω2-ε3ω31-ϱK2A1gS1∫0ω2GK2At-ηgSt-ηK2A1gS1dη+γK1P1gS1∫0ω3GψAt-ηψA1dη.We have U1G(S,L,A,P,Z)>0 for all S,L,A,P,Z>0 and U1G(S1,L1,A1,P1,0)=0. Calculating dU1G/dt, we obtain(54)dU1Gdt=γ1-gS1gSFS-K1P+K2AgS+e-ε3ω3αα+λL1-ξL1ξLϱe-ε1ω1K1Pt-ω1+K2At-ω1gSt-ω1-α+λLξL+e-ε3ω31-ψA1ψA1-ϱe-ε2ω2K1Pt-ω2+K2At-ω2gt-ω2-λψA+αξL+γK1P1gS1nλe-ε3ω3ψA11-φ1P1φ1Pnλe-ε3ω3ψAt-ω3-cφ1P-aφ1Pφ2Z+γqK1P1gS1rnλe-ε3ω3ψA1rφ1Pφ2Z-mφ2Z+e-ε1ω1-ε3ω3αϱα+λLK1P1gS1K1PgSK1P1gS1-K1Pt-ω1gSt-ω1K1P1gS1+ln⁡K1Pt-ω1gSt-ω1K1PgS+e-ε2ω2-ε3ω31-ϱK1P1gS1K1PgSK1P1gS1-K1Pt-ω2gSt-ω2K1P1gS1+ln⁡K1Pt-ω2gSt-ω2K1PgS+e-ε1ω1-ε3ω3αϱα+λLK2A1gS1K2AgSK2A1gS1-K2At-ω1gSt-ω1K2A1gS1+ln⁡K2At-ω1gSt-ω1K2AgS+e-ε2ω2-ε3ω31-ϱK2A1gS1K2AgSK2A1gS1-K2At-ω2gSt-ω2K2A1gS1+lnK2At-ω2gSt-ω2K2AgS+γK1P1gS1ψAψA1-ψAt-ω3ψA1+ln⁡ψAt-ω3ψA.Simplifying (54), we get (55)dU1Gdt=γ1-gS1gSFS+γK1P+K2AgS1-e-ε1ω1-ε3ω3αϱα+λLK1Pt-ω1gSt-ω1ξL1ξL+K2At-ω1gSt-ω1ξL1ξL+αe-ε3ω3ξL1-λe-ε3ω3ψA-e-ε2ω2-ε3ω31-ϱK1Pt-ω2gSt-ω2ψA1ψA+K2At-ω2gSt-ω2ψA1ψA+λe-ε3ω3ψA1-αe-ε3ω3ξLψA1ψA-γK1P1gS1cφ1Pnλe-ε3ω3ψA1+ψAt-ω3φ1P1ψA1φ1P-cφ1P1nλe-ε3ω3ψA1-qφ1P1φ2Znλe-ε3ω3ψA1+qmφ2Zrnλe-ε3ω3ψA1+e-ε1ω1-ε3ω3αϱα+λLK1P1gS1ln⁡K1Pt-ω1gSt-ω1K1PgS+e-ε2ω2-ε3ω31-ϱK1P1gS1ln⁡K1Pt-ω2gSt-ω2K1PgS+e-ε1ω1-ε3ω3αϱα+λLK2A1gS1ln⁡K2At-ω1gSt-ω1K2AgS+e-ε2ω2-ε3ω31-ϱK2A1gS1ln⁡K2At-ω2gSt-ω2K2AgS+γK1P1gS1ln⁡ψAt-ω3ψA+γK1P1gS1ψAψA1.The components of Θ1 satisfy (56)FS1=K1P1+K2A1gS1,e-ε1ω1-ε3ω3αϱα+λLK1P1+K2A1gS1=αe-ε3ω3ξL1,γK1P1+K2A1gS1=λe-ε3ω3ψA1,φ1P1=nλe-ε3ω3ψA1c.We get (57)dU1Gdt=γ1-gS1gSFS-FS1+γK1P1+K2A1gS1-γK1P1+K2A1gS1gS1gS+γK1P+K2AgS1-e-ε1ω1-ε3ω3αϱα+λLK1Pt-ω1gSt-ω1ξL1ξL+K2At-ω1gSt-ω1ξL1ξL+e-ε1ω1-ε3ω3αϱα+λLK1P1+K2A1gS1-γK2A1gS1ψAψA1-e-ε2ω2-ε3ω31-ϱK1Pt-ω2gSt-ω2ψA1ψA+K2At-ω2gSt-ω2ψA1ψA+γK1P1+K2A1gS1-e-ε1ω1-ε3ω3αϱα+λLK1P1+K2A1gS1ξLψA1ξL1ψA-γK1P1gS1φ1Pφ1P1+ψAt-ω3φ1P1ψA1φ1P-1+γqK1P1gS1cφ1P1φ1P1-φ1P2φ2Z+e-ε1ω1-ε3ω3αϱα+λLK1P1gS1ln⁡K1Pt-ω1gSt-ω1K1PgS+e-ε2ω2-ε3ω31-ϱK1P1gS1ln⁡K1Pt-ω2gSt-ω2K1PgS+e-ε1ω1-ε3ω3αϱα+λLK2A1gS1ln⁡K2At-ω1gSt-ω1K2AgS+e-ε2ω2-ε3ω31-ϱK2A1gS1ln⁡K2At-ω2gSt-ω2K2AgS+γK1P1gS1ln⁡ψAt-ω3ψA.Using the equalities in case of i=1,(58)ln⁡K1Pt-ω1gSt-ω1K1PgS=ln⁡K1Pt-ω1gSt-ω1ξLiK1PigSiξL+ln⁡ξLψAiξLiψA+ln⁡ψAK1PiψAiK1P+ln⁡gSigS,ln⁡K1Pt-ω2gSt-ω2K1PgS=ln⁡K1Pt-ω2gSt-ω2ψAiK1PigSiψA+ln⁡ψAK1PiψAiK1P+ln⁡gSigS,ln⁡K2At-ω1gSt-ω1K2AgS=ln⁡K2At-ω1gSt-ω1ξLiK2AigSiξL+ln⁡ξLψAiξLiψA+ln⁡ψAK2AiψAiK2A+ln⁡gSigS,ln⁡K2At-ω2gSt-ω2K2AgS=ln⁡K2At-ω2gSt-ω2ψAiK2AigSiψA+ln⁡ψAK2AiψAiK2A+ln⁡gSigS,ln⁡ψAt-ω3ψA=ln⁡ψAt-ω3φ1PiψAiφ1P+ln⁡ψAiφ1PψAφ1Pi,Therefore, we obtain (59)dU1Gdt=γ1-gS1gSFS-FS1+γK1P1gS1K1P1φ1PK1Pφ1P1+K1PK1P1-1-φ1Pφ1P1+γK2A1gS1ψAK2A1ψA1K2A+K2AK2A1-1-ψAψA1-γK1P1+K2A1gS1gS1gS-1-ln⁡gS1gS-e-ε1ω1-ε3ω3αϱα+λLK1P1gS1K1Pt-ω1gSt-ω1ξL1K1P1gS1ξL-1-ln⁡K1Pt-ω1gSt-ω1ξL1K1P1gS1ξL-e-ε1ω1-ε3ω3αϱα+λLK2A1gS1K2At-ω1gSt-ω1ξL1K2A1gS1ξL-1-ln⁡K2At-ω1gSt-ω1ξL1K2A1gS1ξL-e-ε2ω2-ε3ω31-ϱK1P1gS1K1Pt-ω2gSt-ω2ψA1K1P1gS1ψA-1-ln⁡K1Pt-ω2gSt-ω2ψA1K1P1gS1ψA-e-ε2ω2-ε3ω31-ϱK2A1gS1K2At-ω2gSt-ω2ψA1K2A1gS1ψA-1-lnK2At-ω2gSt-ω2ψA1K2A1gS1ψA-e-ε1ω1-ε3ω3αϱα+λLK1P1+K2A1gS1ξLψA1ξL1ψA-1-ln⁡ξLψA1ξL1ψA-γK1P1gS1ψAt-ω3φ1P1ψA1φ1P-1-ln⁡ψAt-ω3φ1P1ψA1φ1P-γK1P1gS1K1P1φ1PK1Pφ1P1-1-ln⁡K1P1φ1PK1Pφ1P1-γK2A1gS1ψAK2A1ψA1K2A-1-ln⁡ψAK2A1ψA1K2A+γqK1P1gS1cφ1P1φ1P1-φ1P2φ2Z.dU1Gdt=γ1-gS1gSFS-FS1+γK1P1gS1K1PK1P1-φ1Pφ1P11-K1P1K1P+γK2A1gS1K2AK2A1-ψAψA11-K2A1K2A-γK1P1+K2A1gS1GgS1gS-e-ε1ω1-ε3ω3αϱα+λLK1P1gS1GK1Pt-ω1gSt-ω1ξL1K1P1gS1ξL-e-ε1ω1-ε3ω3αϱα+λLK2A1gS1GK2At-ω1gSt-ω1ξL1K2A1gS1ξL-e-ε2ω2-ε3ω31-ϱK1P1gS1GK1Pt-ω2gSt-ω2ψA1K1P1gS1ψA-e-ε2ω2-ε3ω31-ϱK2A1gS1GK2At-ω2gSt-ω2ψA1K2A1gS1ψA-e-ε1ω1-ε3ω3αϱα+λLK1P1+K2A1gS1GξLψA1ξL1ψA-γK1P1gS1GψAt-ω3φ1P1ψA1φ1P-γK1P1gS1GK1P1φ1PK1Pφ1P1-γK2A1gS1GψAK2A1ψA1K2A+γqK1P1gS1cφ1P1φ1P1-φ1P2φ2Z.If R1G≤1, then Θ2 does not exist since Z2=φ2-1(c/q)R1G-1≤0. This means Zt=rφ1P-mφ2Z=rφ1P-φ1P2φ2Z≤0, for all Z>0. Thus, φ1P≤φ1P2 for all P≥0. It follows that φ1P1≤φ1P2. Then according to (H1)-(H3), we have dU1G/dt≤0. The solutions of system (5)-(9) tend to D1G′ the largest invariant subset of D1G=S,L,A,P,Z:dU1G/dt=0. It is easy to show that D1G′=Θ1. Applying LaSalle’s invariance principle, we get that if R1G≤1 and R0G>1, then Θ1 is GAS.On the other hand, it is not hard to verify that the equilibriumΘ2 does not exist if R1G<1, and Θ2=Θ1 if R1G=1. When R1G>1, we have the following result.Theorem 6.
For system (5)-(9), suppose that R1G>1 and let (H1)-(H3) be held true; then, Θ2 is GAS.Proof.
ConsiderU2G(S,L,A,P,Z)(60)U2G=γS-S2-∫S2SgS2gηdη+αe-ε3ω3α+λLL-L2-∫L2LξL2ξηdη+e-ε3ω3A-A2-∫A2AψA2ψηdη+γK1P2gS2nλe-ε3ω3ψA2P-P2-∫P2Pφ1P2φ1ηdη+γqK1P2gS2rnλe-ε3ω3ψA2Z-Z2-∫Z2Zφ2Z2φ2ηdη+e-ε1ω1-ε3ω3αϱα+λLK1P2gS2∫0ω1GK1Pt-ηgSt-ηK1P2gS2dη+e-ε2ω2-ε3ω31-ϱK1P2gS2∫0ω2GK1Pt-ηgSt-ηK1P2gS2dη+e-ε1ω1-ε3ω3αϱα+λLK2A2gS2∫0ω1GK2At-ηgSt-ηK2A2gS2dη+e-ε2ω2-ε3ω31-ϱK2A2gS2∫0ω2GK2At-ηgSt-ηK2A2gS2dη+γK1P2gS2∫0ω3GψAt-ηψA2dη.We have U2G(S,L,A,P,Z)>0 for all S,L,A,P,Z>0, while U2G reaches its global minimum at Θ2. Calculate dU2G/dt as(61)dU2Gdt=γ1-gS2gSFS-K1P+K2AgS+e-ε3ω3αα+λL1-ξL2ξLϱe-ε1ω1K1Pt-ω1+K2At-ω1gSt-ω1-α+λLξL+e-ε3ω31-ψA2ψA1-ϱe-ε2ω2K1Pt-ω2+K2At-ω2gSt-ω2-λψA+αξL+γK1P2gS2nλe-ε3ω3ψA21-φ1P2φ1Pnλe-ε3ω3ψAt-ω3-cφ1P-qφ1Pφ2Z+γqK1P2gS2rnλe-ε3ω3ψA21-φ2Z2φ2Zrφ1Pφ2Z-mφ2Z+e-ε1ω1-ε3ω3αϱα+λLK1P2gS2K1PgSK1P2gS2-K1Pt-ω1gSt-ω1K1P2gS2+ln⁡K1Pt-ω1gSt-ω1K1PgS+e-ε2ω2-ε3ω31-ϱK1P2gS2K1PgSK1P2gS2-K1Pt-ω2gSt-ω2K1P2gS2+ln⁡K1Pt-ω2gSt-ω2K1PgS+e-ε1ω1-ε3ω3αϱα+λLK2A2gS2K2AgSK2A2gS2-K2At-ω1gSt-ω1K2A2gS2+ln⁡K2At-ω1gSt-ω1K2AgS+e-ε2ω2-ε3ω31-ϱK2A2gS2K2AgSK2A2gS2-K2At-ω2gSt-ω2K2A2gS2+ln⁡K2At-ω2gSt-ω2K2AgS+γK1P2gS2ψAψA2-ψAt-ω3ψA2+ln⁡ψAt-ω3ψA.Simplify (61) as (62)dU2Gdt=γ1-gS2gSFS+γK1P+K2AgS2-e-ε1ω1-ε3ω3αϱα+λLK1Pt-ω1gSt-ω1ξL2ξL+K2At-ω1gSt-ω1ξL2ξL+αe-ε3ω3ξL2-λe-ε3ω3ψA-e-ε2ω2-ε3ω31-ϱK1Pt-ω2gSt-ω2ψA2ψA+K2At-ω2gSt-ω2ψA2ψA+λe-ε3ω3ψA2-αe-ε3ω3ξLψA2ψA-γK1P2gS2cφ1Pnλe-ε3ω3ψA2+ψAt-ω3φ1P2ψA2φ1P-cφ1P2nλe-ε3ω3ψA2-qφ1P2φ2Znλe-ε3ω3ψA2+qmφ2Zrnλe-ε3ω3ψA2+qφ1Pφ2Z2nλe-ε3ω3ψA2-qmφ2Z2rnλe-ε3ω3ψA2+e-ε1ω1-ε3ω3αϱα+λLK1P2gS2ln⁡K1Pt-ω1gSt-ω1K1PgS+e-ε2ω2-ε3ω31-ϱK1P2gS2ln⁡K1Pt-ω2gSt-ω2K1PgS+e-ε1ω1-ε3ω3αϱα+λLK2A2gS2ln⁡K2At-ω1gSt-ω1K2AgS+e-ε2ω2-ε3ω31-ϱK2A2gS2ln⁡K2At-ω2gSt-ω2K2AgS+γK1P2gS2ln⁡ψAt-ω3ψA+γK1P2gS2ψAψA2.The components of Θ2 satisfy(63)FS2=K1P2+K2A2gS2,e-ε1ω1-ε3ω3αϱα+λLK1P2+K2A2gS2=αe-ε3ω3ξL2,γK1P2+K2A2gS2=λe-ε3ω3ψA2,nλe-ε3ω3ψA2=cφ1P2+qφ1P2φ2Z2,φ1P2=mr,we get(64)dU2Gdt=γ1-gS2gSFS-FS2+γK1P2+K2A2gS2-γK1P2+K2A2gS2gS2gS+γK1P+K2AgS2-e-ε1ω1-ε3ω3αϱα+λLK1Pt-ω1gSt-ω1ξL2ξL+K2At-ω1gSt-ω1ξL2ξL+e-ε1ω1-ε3ω3αϱα+λLK1P2+K2A2gS2-γK2A2gS2ψAψA2-e-ε2ω2-ε3ω31-ϱK1Pt-ω2gSt-ω2ψA2ψA+K2At-ω2gSt-ω2ψA2ψA+e-ε1ω1-ε3ω3αϱα+λLK1P2+K2A2gS2+e-ε2ω2-ε3ω31-ϱK1P2+K2A2gS2-e-ε1ω1-ε3ω3αϱα+λLK1P2+K2A2gS2ξLψA2ξL2ψA-γK1P2gS2φ1Pφ1P2+ψAt-ω3φ1P2ψA2φ1P-1+e-ε1ω1-ε3ω3αϱα+λLK1P2gS2ln⁡K1Pt-ω1gSt-ω1K1PgS+e-ε2ω2-ε3ω31-ϱK1P2gS2ln⁡K1Pt-ω2gSt-ω2K1PgS+e-ε1ω1-ε3ω3αϱα+λLK2A2gS2ln⁡K2At-ω1gSt-ω1K2AgS+e-ε2ω2-ε3ω31-ϱK2A2gS2ln⁡K2At-ω2gSt-ω2K2AgS+γK1P2gS2ln⁡ψAt-ω3ψA.Using equalities (58) when i=2, we obtain (65)dU2Gdt=γ1-gS2gSFS-FS2+γK1P2gS2K1P2φ1PK1Pφ1P2+K1PK1P2-1-φ1Pφ1P2+γK2A2gS2K2A2ψAK2AψA2+K2AK2A2-1-ψAψA2-γK1P2+K2A2gS2gS2gS-1-ln⁡gS2gS-e-ε1ω1-ε3ω3αϱα+λLK1P2gS2K1Pt-ω1gSt-ω1ξL2K1P2gS2ξL-1-ln⁡K1Pt-ω1gSt-ω1ξL2K1P2gS2ξL-e-ε1ω1-ε3ω3αϱα+λLK2A2gS2K2At-ω1gSt-ω1ξL2K2A2gS2ξL-1-ln⁡K2At-ω1gSt-ω1ξL2K2A2gS2ξL-e-ε2ω2-ε3ω31-ϱK1P2gS2K1Pt-ω2gSt-ω2ψA2K1P2gS2ψA-1-ln⁡K1Pt-ω2gSt-ω2ψA2K1P2gS2ψA-e-ε2ω2-ε3ω31-ϱK2A2gS2K2At-ω2gSt-ω2ψA2K2A2gS2ψA-1-ln⁡K2At-ω2gSt-ω2ψA2K2A2gS2ψA-e-ε1ω1-ε3ω3αϱα+λLK1P2+K2A2gS2ξLψA2ξL2ψA-1-ln⁡ξLψA2ξL2ψA-γK1P2gS2ψAt-ω3φ1P2ψA2φ1P-1-ln⁡ψAt-ω3φ1P2ψA2φ1P-γK1P2gS2K1P2φ1PK1Pφ1P2-1-ln⁡K1P2φ1PK1Pφ1P2-γK2A2gS2K2A2ψAK2AψA2-1-ln⁡K2A2ψAK2AψA2.dU2Gdt=γ1-gS2gSFS-FS2+γK1P2gS2K1PK1P2-φ1Pφ1P21-K1P2K1P+γK2A2gS2K2AK2A2-ψAψA21-K2A2K2A-γK1P2+K2A2gS2GgS2gS-e-ε1ω1-ε3ω3αϱα+λLK1P2gS2GK1Pt-ω1gSt-ω1ξL2K1P2gS2ξL-e-ε1ω1-ε3ω3αϱα+λLK2A2gS2GK2At-ω1gSt-ω1ξL2K2A2gS2ξL-e-ε2ω2-ε3ω31-ϱK1P2gS2GK1Pt-ω2gSt-ω2ψA2K1P2gS2ψA-e-ε2ω2-ε3ω31-ϱK2A2gS2GK2At-ω2gSt-ω2ψA2K2A2gS2ψA-e-ε1ω1-ε3ω3αϱα+λLK1P2+K2A2gS2GξLψA2ξL2ψA-γK1P2gS2GψAt-ω3φ1P2ψA2φ1P-γK1P2gS2GK1P2φ1PK1Pφ1P2-γK2A2gS2GK2A2ψAK2AψA2.From (H1)-(H3), we have that dU2G/dt≤0 for all S,L,A,P and Z>0. One can show that dU2G/dt=0 at Θ2; then apply LaSalle’s invariance principle to show the global asymptotic stability of Θ2.


## 2. Numerical Simulations

For numerical simulation purpose, we consider model (5)-(9) with the following specific functions:(66)FS=ϰ-dS,gS=S1+γHS,K1P=b1P,K2A=b2A,ξL=L,ψA=A,φ1P=P,φ2Z=Z.First, we show that (H1)-(H3) are satisfied. We have S0=ϰ/d, FS0=0, F0=ϰ>0, and F′S=-d<0. It follows that FS>0 for all S∈[0,S0). Moreover, (H1)(iii) is satisfied with z=ϰ and z¯=d. Thus, (H1) is satisfied. We have gS,K1P,K2A,ξL,ψA,φ1P,φ2Z>0 for all S,P,A,L,Z>0 and g0=K10=K20=ξ0=ψ0=φ10=φ20=0. We also have(67)g′S=11+γHS2>0∀S>0,K1′P=b1>0∀P≥0,K2′A=b2>0∀A≥0,ξ′L=1>0∀L≥0,ψ′A=1>0∀A≥0,φ1′P=1>0∀P≥0,φ2′Z=1>0∀Z≥0.Hence, (H2) (i)-(ii) are satisfied. Moreover, (H2) (iii) is satisfied, where c1=c2=c3=c4=1. Finally, we have(68)K1Pφ1P′=0∀P>0,K2AψA′=0∀A>0.Thus, (H3) is also valid.With the above choice, system (5)-(9) will be(69)S˙t=ϰ-dSt-b1StPt1+γHSt-b2StAt1+γHSt,(70)L˙t=ϱe-ε1ω1b1St-ω1Pt-ω11+γHSt-ω1+b2St-ω1At-ω11+γHSt-ω1-α+λLLt,(71)A˙t=1-ϱe-ε2ω2b1St-ω2Pt-ω21+γHSt-ω2+b2St-ω2At-ω21+γHSt-ω2-λAt+αLt,(72)P˙t=nλe-ε3ω3At-ω3-cPt-qPtZt,(73)Z˙t=rPtZt-mZt,where γH is the Holling-type II constant. The numbers R0G and R1G are given as(74)R0G=γS0nλe-ε3ω3b1+cb2cλe-ε3ω31+γHS0,(75)R1G=rnλe-ε3ω3cmA2.Therefore, Theorems3–6 are valid for such choice.Let us considerω=ω1=ω2=ω3 and the following initial values:I V 1: χ1(η)=800, χ2(η)=2, χ3(η)=2, χ4(η)=6, χ5(η)=2,I V 2: χ1(η)=600, χ2(η)=4, χ3(η)=6, χ4(η)=9, χ5(η)=4,I V 3: χ1(η)=400, χ2(η)=6, χ3(η)=9, χ4(η)=15, χ5(η)=6,I V 4: χ1(η)=700, χ2(η)=1, χ3(η)=1, χ4(η)=9, χ5(η)=4, η∈-ω,0.The stability of the equilibria will be investigated by varying some parametersr,γH,b1,b2, and ω and fixing the other parameters as ϰ=10, d=0.01, λ=λu=0.5, c=2, ϱ=0.5, α=0.05, n=10, q=0.1, m=0.1, and ε1=ε2=ε3=1.Case (I) (effect of the parameters b1,b2, and r). We take γH=0, ω=0. We choose three sets of the parameters b1, b2, and r to confirm Theorems 3–6 (see Figure 1).Figure 1
The evolution of system (69)-(73) for Case (I).
(a) (b) (c) (d) (e)Set (  1) (stability of Θ0). b1=0.0001, b2=0.0001, and r=0.001. This yields R0G=0.3818<1 and R1G=0.1401<1.Set (  2) (stability of Θ1). b1=0.0006, b2=0.0006, and r=0.001. With these values, we obtain R0G=2.2909>1 and R1G=0.2367<1.Set (  3) (stability of Θ2). b1=0.0006, b2=0.0006, and r=0.04 and then R0G=2.2909>1 and R1G=2.5285>1.Case (II) (effect of the Holling-type II parameter). Let us take IV1 and choose the values b1=0.0006, b2=0.0006, ω=0, and r=0.04. Figure 2 and Table 1 show the effect of Holling-type II parameter γH on the stability of the system. We observe that as γH is increased, the pathogen-to-susceptible and infected-to-susceptible transmission rates are decreased. Then, the concentrations of the susceptible cells are increased, while the concentrations of the infected cells and free pathogens are decreased. In addition,Table 1
The values ofR0G and R1G for system (69)-(73) with different values of γH.
γ H Equilibrium R 0 G R 1 G 0.0 Θ 2 2.29091 2.52851 0.0007 Θ 2 1.34759 1.37541 0.0009 Θ 2 1.20574 1.19569 0.00118999 Θ 1 1.04608 1 0.00125 Θ 1 1.01818 0.966725 0.00129091 Θ 0 1 0.945205 0.05 Θ 0 0.0449198 0.0325008 1.0 Θ 0 0.00228862 0.0016358Figure 2
The evolution of system (69)-(73) with different values of γH for Case (II).
(a) (b) (c) (d) (e)(i)Θ2 is GAS when 0≤γH<0.00099,(ii)Θ1 is GAS when 0.00099≤γH<0.00129,(iii)Θ0 is GAS when γH≥0.00129.Case (III) (effect of time delay parameters). For this case, we take IV4 and choose the values b1=0.0006, b2=0.0006, γH=0.00035, and r=0.04. Figure 3 and Table 2 show the effect of the time delay on the dynamical behavior of the pathogen. We have the following cases:Table 2
The values ofR0G and R1G for system (69)-(73) with different values of ω.
ω Equilibrium R 0 G R 1 G 0.0 Θ 2 = (832.667, 1.52121, 1.82545, 2.5, 16.5091) 1.69697 1.82545 0.1 Θ 2 = (841.463, 1.3041, 1.56491, 2.5, 8.31986) 1.43111 1.41599 0.2412 Θ 1 = (851.503, 1.06065, 1.27278, 2.50001, 0) 1.12918 1 0.29 Θ 1 = (946.997, 0.360549, 0.432659, 0.809357, 0) 1.04146 0.889019 0.314637 Θ 0 = (1000, 0, 0, 0, 0) 1 0.838138 0.5 Θ 0 = (1000, 0, 0, 0, 0) 0.73999 0.542645 1 Θ 0 = (1000, 0, 0, 0, 0) 0.342409 0.177726Figure 3
The evolution of system (69)-(73) with different values of ω for Case (III).
(a) (b) (c) (d) (e)(i)Θ2 is GAS when 0≤ω<0.2412,(ii)Θ1 is GAS when 0.2412≤ω<0.314637,(iii)Θ0 is GAS when ω≥0.314637.Therefore, if the time delay is long enough, then the system can be stabilized aroundΘ0, where the pathogens are cleared



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