# Stationary Distribution and Extinction of a Stochastic Viral Infection Model

Authors: Yan Wang; Daqing Jiang
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

We present a kind of stochastic viral infection model with or without a loss term in the free virus equation. We obtain critical condition to ensure the existence of the unique stationary distribution by constructing Lyapunov functions. We also obtain the sufficient conditions for the extinction of the virus by the comparison theorem of stochastic differential equation and law of large numbers. We give a unified method to systematically analyze such three-dimensional stochastic viral infection model. Furthermore, numerical simulations are carried out to examine the effect of white noises on model behavior. We investigate the fact that the small magnitudes of white noises can sustain the irregular recurrence of healthy target cells and virions, while the big ones may contribute to viral clearance.


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## Body

## 1. Introduction

More and more attention has been paid to the mathematical modelling of virus (such as HIV and hepatitis B/C virus) infection. The existing viral models mainly contain three compartments: target cells, actively infected cells, and matured virions; see, for example, [1–10] and the references cited therein. These models are all derived from the basic viral dynamics models [1–3]. However, all these viral models are deterministic models and do not take into account the stochastic fluctuation. Previously, through experimental data, Singh et al. have investigated the fact that the presence of random noise in gene expression may lead to an increase of variability in HIV early gene products, which may seriously affect the fate of the virus between replication and latency [11]. Thus, Mao et al. have shown that, from the perspective of mathematical analysis, a small amount of random fluctuation can inhibit a potential population explosion [12]. Hence, including the effect of random noise in viral dynamics model may play an important role in the development of a better understanding of the disease.With the development of stochastic epidemic models of between-host transmission [13–17], the stochastic models of within-host viral infection have also been developed in recent years [18–23]. For most of models, these have been considered as the stochastic effects of the model parameters [19–21, 23], but these models did not consider the loss term of virus when a free virus enters the target cell. This term is often neglected even though in the deterministic models, because it is typically very small and can be absorbed into the virus clearance term. However, recent studies have showed that the inclusion of this term may have a major impact on the basic reproductive number and viral dynamics under certain parameter sets [4, 5, 24]. It may also affect the probability of extinction of an initial viral load through Monte Carlo approaches [25, 26]. Therefore, it is more reasonable to include the virus loss term in stochastic viral infection model, which would have further influence on the model behavior. Few stochastic viral infection models [18, 22] have included the virus loss term, while the theoretical results are needed to be complete.In this paper, we extend the classical deterministic viral infection model with or without the virus loss term to a stochastic model of parameter fluctuations. The rest of this paper is organized as follows. In the next section, we formulate our model and show the existence and uniqueness of the global positive solution. In Section3, by constructing suitable Lyapunov functions, the existence of a unique ergodic stationary distribution is derived. We establish the sufficient conditions for the extinction of the virus in Section 4. In Section 5, by employing numerical simulations, we investigate how the white noises affect the model dynamic behavior under realistic parameters. Finally, we conclude our results and give future work.


## 2. Model Derivation

The basic viral infection model [1–3] with or without a loss term in free virus equation can be described by the following ordinary differential equations:(1)dTdt=λ-dT-kTV,dT∗dt=kTV-δT∗,dVdt=NδT∗-cV-ikTV,i=0or1,where T, T∗, and V denote the concentrations of healthy target cells, actively infected target cells, and infectious virus particles, respectively. Parameter λ is the constant input rate, k presents the viral infection rate between healthy target cells and infectious virus. Parameters d and δ are the death rates of the healthy and infected target cells, respectively. N is the average number of virus particles produced by an infected target cell and c is the clearance rate of the infectious virus. To reflect the model inclusion or omission of a loss term in free virus equation, we use a term -ikVT in the third equation and i is a fixed constant. When i=0, it means that model (1) ignores the loss term kVT due to infection; when i=1, it means that model (1) includes the loss term kVT due to infection.In literature [4], de Leenheer and Smith have obtained the theoretical results of (1). System (1) always has a disease-free steady state E0(T0,0,0), where T0=λ/d. The basic reproduction number is given by (2)R0,i=kNT0c+ikT0,i=0or1.System (1) has the following main results: (i)
IfR0<1, the disease-free steady state E0 is globally asymptotically stable.(ii)
IfR0>1, then a chronic disease steady state exists which is globally asymptotically stable, and the disease-free steady state E0 is unstable.In this paper, considering random fluctuation, we assume that the stochastic fluctuation is the white noise type, which is a linear perturbation corresponding to the rate of change for each population. The stochastic differential model can be written as follows:(3)dT=λ-dT-kTVdt+σ1TdB1t,dT∗=kTV-δT∗dt+σ2T∗dB2t,dV=NδT∗-cV-ikTVdt+σ3VdB3t,i=0or1,where Bi(t) are independent standard Brownian motions with Bi(0)=0 and σi2>0 denote the intensities of the white noise, i=1,2,3. The other parameters are the same as those in system (1).In this paper, we assume(Ω,F,{Ft}t≥0,P) as a complete probability space with a filtration {Ft}t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contains all P-null sets), and Bi(t)(i=1,2,3) are defined on this complete probability space. We also let R+3={x=(x1,x2,x3)∈R3:xi>0,i=1,2,3}. We use a∧b to denote min⁡{a,b} and use a∨b to denote max⁡{a,b}.According to the theory of Arnold [27] and Mao et al. [12], we show the existence and uniqueness of the solution of system (3) in the following Lemma. See Theorem  4.1 of [18] for the detailed proof.Lemma 1.
System (3) has a unique and positive solution (T(t),T∗(t),V(t)) with the initial value (T(0),T∗(0),V(0)) for all t≥0, and the solution will remain in R+3 with probability one, namely, (T(t),T∗(t),V(t))∈R+3  for all t≥0 almost surely (a.s.).


## 3. Stationary Distribution

In this section, by using the theory of Hasminskii [28], we prove the existence of a unique ergodic stationary distribution, which indicates that the virus is prevalent.Denote(4)σ2=σ12∨σ22∨σ32, and define the critical condition(5)R0,is=λkNδd+1/2σ12δ+1/2σ22c+ikT0+1/2σ32,i=0or1.Theorem 2.
IfR0,is>1, then system (3) has a unique stationary distribution μ(·) and it has the ergodic property.Proof.
We firstly verified the establishment of condition (H1) in LemmaA.1 in Appendix. For system (3), the diffusion matrix is (6)A=σ12T2000σ22T∗2000σ32V2. Choosing E=min(T,T∗,V)∈D¯k⊂R+3{σ12T2,σ22(T∗)2,σ32V2}, we have (7)∑i,j=13aijT,T∗,Vξiξj=σ12T2ξ12+σ22T∗2ξ22+σ32V2ξ32≥Eξ2,T,T∗,V∈D¯k,ξ=ξ1,ξ2,ξ3∈R+3, where D¯k=[1/k,k]×[1/k,k]×[1/k,k], and k is a sufficiently large integer; then condition (H1) in Lemma A.1 is satisfied.
Next, we show the validity of condition (H2) in LemmaA.1 in Appendix. Now, construct a C2-function W1: R+3→R,(8)W1=-ln⁡T-c1ln⁡T∗-c2ln⁡V-ikTd, where i=0 or 1; c1 and c2 are positive constants which will be determined later. Applying Itô’s formula [29] to W1 yields (9)L-ln⁡V+ikTd=-NδT∗V+c+ikλd-ik2VTd+12σ32≤-NδT∗V+c+ikT0+12σ32, and we should mention that T0=λ/d. Thus, (10)LW1≤-λT-c1kTVT∗-c2NδT∗V+d+12σ12+c1δ+12σ22+c2c+ikT0+12σ32+kV≤-3λkNδc1c23+c1δ+12σ22+c2c+ikT0+12σ32+d+12σ12+kV. Let(11)c1δ+12σ22=c2c+ikT0+12σ32=λkNδδ+1/2σ22c+ikT0+1/2σ32, and calculate that(12)c1=λkNδδ+1/2σ222c+ikT0+1/2σ32,c2=λkNδδ+1/2σ22c+ikT0+1/2σ322. Hence,(13)LW1≤-λkNδδ+1/2σ22c+ikT0+1/2σ32+d+12σ12+kV≤-d+12σ12R0,is-1+kV=-β+kV, where(14)R0,is=λkNδd+1/2σ12δ+1/2σ22c+1/2σ32,β=d+12σ12R0,is-1.
Define aC2-function W:R+3→R, in the following form:(15)WT,T∗,V=MW1+W2+W3+W4, where(16)W2=-ln⁡T,W3=-ln⁡T∗,W4=11+θT+T∗+V2N1+θ, where 0<θ<min⁡1,1/2σ2(2d∧δ∧2c). Select a suitable constant M>0 satisfying the following condition: (17)-Mβ+C≤-2, where (18)C=supT,T∗,V∈R+3-d2T1+θ-δ4T∗1+θ-c2V2N1+θ+E+d+δ+12σ12+12σ22<∞. It is easy to check that(19)liminfn→∞,T,T∗,V∈R+3∖DkWT,T∗,V=+∞, where Dk=(1/k,k)×(1/k,k)×(1/k,k). Furthermore, W(T,T∗,V) is a continuous function. Hence W(T,T∗,V) must have a minimum point (T¯0,T∗¯0,V¯0) in the interior of R+3. Then we define a nonnegative C2-function W¯:R+3→R as follows:(20)W¯T,T∗,V=WT,T∗,V-WT¯0,T∗¯0,V¯0. Applying Itô formula, (21)LW2=-λT+d+kV+12σ12,LW3=-kTVT∗+δ+12σ22,LW4=T+T∗+V2Nθλ-dT-δT∗2-cV2N-ikTV2N+θ2T+T∗+V2Nθ-1σ12T2+σ22T∗2+σ32V2N2≤T+T∗+V2Nθλ-dT-δT∗2-cV2N+θ2σ2T+T∗+V2Nθ-1T2+T∗2+V2N2≤λT+T∗+V2Nθ-dT1+θ-δ2T∗1+θ-cV2N1+θ+θ2σ2T1+θ+T∗1+θ+V2N1+θ≤-d2T1+θ-δ4T∗1+θ-c2V2N1+θ+E, where (22)E=supT,T∗,V∈R+3-12d-θσ2T1+θ-14δ-2θσ2T∗1+θ-12c-θσ2V2N1+θ+λT+T∗+V2Nθ<∞. Hence, (23)LW¯≤-Mβ+M+1kV-λT-kTVT∗-d2T1+θ-δ4T∗1+θ-c2V2N1+θ+E+d+δ+12σ12+12σ22.
Next, we construct a compact subsetDε such that condition (H2) in Lemma A.1 holds. Define the following bounded closed set:(24)Dε=ε≤T≤1ε,ε3≤T∗≤1ε3,ε≤V≤1ε, where ε is a sufficiently small constant. In the set R+3∖Dε, we further choose ε sufficiently small such that the following conditions hold:(25)-Mβ+M+1kε+C≤-1,(26)-λε+H≤-1,(27)-kε+H≤-1,(28)-c42Nε1+θ+H≤-1,(29)-d4ε1+θ+H≤-1,(30)-δ8ε31+θ+H≤-1,where (31)H=supT,T∗,V∈R+3M+1kV-d4T1+θ-δ8T∗1+θ-c4V2N1+θ+E+d+δ+12σ12+12σ22<∞. In the following, we separate R+3∖Dε to six domains: (32)D1=T,T∗,V∈R+3,0<V<ε,D2=T,T∗,V∈R+3,0<T<ε,D3=T,T∗,V∈R+3,0<T∗<ε3,T≥ε,V≥ε,D4=T,T∗,V∈R+3,V>1ε,D5=T,T∗,V∈R+3,T>1ε,D6=T,T∗,V∈R+3,T∗>1ε3. Obviously, Dεc=D1∪D2∪D3∪D4∪D5∪D6.
Case  1. If (T,T∗,V)∈D1,(33)LW¯≤-Mβ+M+1kV-d2T1+θ-δ4T∗1+θ-c2V2N1+θ+E+d+δ+12σ12+12σ22≤-Mβ+M+1kε+C. According to (25), it implies that LW¯≤-1 for any (T,T∗,V)∈D1.
Case  2. If (T,T∗,V)∈D2,(34)LW¯≤-λT+M+1kV-d2T1+θ-δ4T∗1+θ-c2V2N1+θ+E+d+δ+12σ12+12σ22≤-λT+H≤-λε+H. In view of (26), we have LW¯≤-1 for any (T,T∗,V)∈D2.
Case  3. If (T,T∗,V)∈D3,(35)LW¯≤-kTVT∗+M+1kV-d2T1+θ-δ4T∗1+θ-c2V2N1+θ+E+d+δ+12σ12+12σ22≤-kVTT∗+H≤-kε+H. According to (27), we deduce that LW¯≤-1 for any (T,T∗,V)∈D3.
Case  4. If (T,T∗,V)∈D4,(36)LW¯≤-c2V2N1+θ+M+1kV-d2T1+θ-δ4T∗1+θ+E+d+δ+12σ12+12σ22≤-c4V2N1+θ+H≤-c42Nε1+θ+H. In view of (30), we obtain that LW¯≤-1 for any (T,T∗,V)∈D4.
Case  5. If (T,T∗,V)∈D4,(37)LW¯≤-d2T1+θ+M+1kV-δ4T∗1+θ-c2V2N1+θ+E+d+δ+12σ12+12σ22≤-d4T1+θ+H≤-d4ε1+θ+H. By condition (28), we can conclude that LW¯≤-1 for all (T,T∗,V)∈D5.
Case  6. If (T,T∗,V)∈D5,(38)LW¯≤-δ4T∗1+θ+M+1kV-d2T1+θ-c2V2N1+θ+E+d+δ+12σ12+12σ22≤-δ8T∗1+θ+H≤-δ8ε31+θ+H. It follows that LW¯≤-1 for any (T,T∗,V)∈D6 if the condition (29) is satisfied.
Obviously, from (33)–(38), there exists a sufficiently small ε, such that(39)LW¯≤-1∀T,T∗,V∈Dεc. According to Lemma A.1 in Appendix, we obtain that system (3) is ergodic and has a unique stationary distribution. This completes the proof.Remark 3.
Theorem2 indicates that the virus prevails if the critical condition R0,is is greater than one. We should mention that the critical condition R0,is of the stochastic model (3) without white noises is consistent with the basic reproduction number R0,i of its corresponding deterministic model (1). This implies that the existence of the stationary distribution of our model is a generalization of its corresponding deterministic model to the stability of the chronic disease steady state.


## 4. Extinction

In this section, following the method in Theorem  3.1 of [30], we employ the comparison theorem of stochastic differential equation [31] and the law of large numbers [32] (see Appendix for details) to derive the sufficient conditions for extinction for the virus.Theorem 4.
Let(T(t),T∗(t),V(t)) be the solution of system (3) with any initial value (T(0),T∗(0),V(0))∈R+3. If d>1/2σ12, then, for almost ω∈Ω, the solution (T(t),T∗(t),V(t)) of system (3) satisfies(40)limsupt→∞1tln⁡Nc+ikT0T∗t+R0,ic+ikT0Vt≤m,i=0or1,a.s., where m=min⁡δ,c+ikT0(R0,i-1)IR0,i≤1+max⁡{δ,c+ikT0}(R0,i-1)I{R0,i>1}+ikT0+kNT0σ1[R0,i(2d-σ12)]-1/2-1/2(σ2-2+σ3-2)-1 and the distribution of T(t) converges weakly to the measure which has the density given by(41)πx=Qσ1-2x-2+2d/σ12e-2λ/σ12x,x∈0,∞, where Q=σ122λ/σ121+2d/σ12Γ-1(2d/σ12+1) such that ∫0∞π(x)dx=1. If m<0, then the populations T∗ and V will be eradicated with probability one; that is,(42)limt→∞T∗t=0,limt→∞Vt=0,a.s.Proof.
By Lemma1, we know that the solution of system (3) is positive, and it is obvious that(43)dT≤λ-dTdt+σ1TdB1t. Consider the following auxiliary equation with stochastic differential equation:(44)dx=λ-dxdt+σ1xdB1t,with the initial value x(0)=T(0)>0. Corresponding to (A.4) in Appendix, this yields(45)gx=λ-dx,σx=σ1x,x∈0,∞, and we further calculate that (46)∫c1xguσ2udu=1σ12∫c1xλu2-dudu=-1σ12λx+dln⁡x+C1, where c1 is an arbitrary fixed positive constant and C1 is a constant determined by the above calculation. It is obvious that (47)∫0∞1σ2xexp2∫c1xguσ2ududx=Qσ1-2∫0∞x-2+2d/σ12e-2λ/σ12xdx<∞, where Q=e2C1. By Lemma A.2 in Appendix, we obtain that system (44) has the ergodic property with ergodic distribution:(48)πx=Qσ1-2x-2+2d/σ12e-2λ/σ12x,x∈0,∞, where Q=σ122λ/σ121+2d/σ12Γ-1(2d/σ12+1) such that ∫0∞π(x)dx=1. It then follows that (49)limt→∞1t∫0txsds=∫0∞xπxdxa.s.
Letx(t) be the solution of system (44) with the initial value x(0)=T(0)>0, by the comparison theorem of stochastic differential equation [31], we obtain that (50)Tt≤xta.s.
We further calculate that(51)A1≔∫0∞xπxdx=Qσ1-2∫0∞x-1+2d/σ12e-2λ/σ12xdx=Qσ1-22λσ12-2d/σ12∫0∞u2d/σ12-1e-udu=Qσ1-22λσ12-2d/σ12Γ2dσ12=2λσ12Γ2d/σ12Γ2d/σ12+1=λd,A2≔∫0∞x2πxdx=Qσ1-2∫0∞x-2d/σ12e-2λ/σ12xdx=Qσ1-22λσ121-2d/σ12∫0∞u2d/σ12-2e-udu=Qσ1-22λσ121-2d/σ12Γ2dσ12-1=2λσ122Γ2d/σ12-1Γ2d/σ12+1=2λ2d2d-σ12. Therefore, (52)∫0∞x-T02πxdx=∫0∞x2πxdx-T0∫0∞xπxdx+T02=A2-2T0A1+T02=2λ2d2d-σ12-2λd2+λd2=λ2σ12d22d-σ12.
Denote the matrix(53)F=0kT0δNδc+ikT00. By Theorem  1.4 of literature [33], we know that there must be a left eigenvector (ω1,ω2) of matrix F corresponding to the value R0,i, such that(54)ω1,ω2F=R0,iω1,ω2, where (ω1,ω2)=Nδ/(c+ikT0),R0,i.
Define aC2-function W:R+3→R:(55)GT∗,V=α1T∗+α2V, where α1=ω1/δ,α2=ω2/(c+ikT0). Applying Itô’s formula yields(56)dln⁡G=Lln⁡Gdt+1Gα1σ2T∗dB2t+α2σ3VdB3t,where (57)Lln⁡G=α1GkTV-δT∗+α2GNδT∗-cV-ikTV-α12σ22T∗22G2-α22σ32V22G2≔B-α12σ22T∗22G2-α22σ32V22G2. Furthermore, we have(58)G2=α1σ2T∗1σ2+α2σ3V1σ32≤α12σ22T∗2+α22σ32V21σ22+1σ32,B=1Gα1kTV-δT∗+α2NδT∗-cV-ikTV=1Gα1kT-T0V+kT0V-δT∗+α2NδT∗-cV-ikT-T0V-ikT0V=α1kVGT-T0+1Gα1kT0V-δT∗+α2NδT∗-cV-ikT0V-α2GikT-T0V≤α1kα2T-T0+1Gω1δkT0V-δT∗+ω2c+ikT0NδT∗-cV-ikT0V+ikT0=α1kα2T-T0+1Gω1,ω2FT∗,VT-T∗,VT+ikT0=α1kα2T-T0+ikT0+1GR0,i-1ω1T∗+ω2V=α1kα2T-T0+ikT0+1GR0,i-1α1δT∗+c+ikT0α2V≤α1kα2T-T0+ikT0+min⁡δ,c+ikT0R0,i-1IR0,i≤1+max⁡δ,c+ikT0R0,i-1IR0,i>1. According to (58), we get that (59)Lln⁡G≤min⁡δ,c+ikT0R0,i-1IR0,i≤1+max⁡δ,c+ikT0R0,i-1IR0,i>1+α1kα2T-T0+ikT0-12σ2-2+σ3-2-1. From (56), we further obtain that(60)dln⁡G≤min⁡δ,c+ikT0R0,i-1IR0,i≤1+max⁡δ,c+ikT0×R0,i-1IR0,i>1+α1kα2T-T0+ikT0-12σ2-2+σ3-2-1dt+1Gα1σ2T∗dB2t+α2σ3VdB3t. Integrating (60) from 0 to t, then dividing by t on both sides, yields(61)ln⁡Gtt≤ln⁡G0t+min⁡δ,c+ikT0R0,i-1IR0,i≤1+max⁡δ,c+ikT0×R0,i-1IR0,i>1+α1kα2t∫0tTs-T0ds+ikT0-12σ2-2+σ3-2-1+1t∫0tα1σ2T∗sGsdB2s+1t∫0tα2σ3VsGsdB3s=ln⁡G0t+min⁡δ,c+ikT0R0,i-1IR0,i≤1+max⁡δ,c+ikT0×R0,i-1IR0,i>1+α1kα2t∫0tTs-T0ds+ikT0-12σ2-2+σ3-2-1+M1tt+M2tt, where (62)M1t=∫0tα1σ2T∗sGsdB2s,M2t=∫0tα2σ3VsGsdB3s.M1(t) and M2(t) are local martingales whose quadratic variations are M1,M1t=σ22×∫0tα1T∗(s)/G(s)ds≤σ22t and M2,M2t=σ32∫0tα2V(s)/G(s)ds≤σ32t. Applying the strong law of large numbers for local martingale [29] yields(63)limt→∞Mitt=0a.s.,i=1,2.Since T(t) is ergodic and ∫0∞xπ(x)dx<∞, so(64)limt→∞1t∫0tTs-T0ds=∫0∞x-T0πxdx≤∫0∞x-T02πxdx1/2=T0σ12d-σ12-1/2. Taking the superior limit on both sides of inequality (61) and combining with (63) and (64), we obtain (65)limsupt→∞ln⁡Gtt≤min⁡δ,c+ikT0R0,i-1IR0,i≤1+max⁡δ,c+ikT0×R0,i-1IR0,i>1+α1kα2t∫0tTs-T0ds+ikT0-12σ2-2+σ3-2-1≤min⁡δ,c+ikT0R0,i-1IR0,i≤1+max⁡δ,c+ikT0×R0,i-1IR0,i>1+ikT0+kNT0σ1R0,i2d-σ12-1/2-12σ2-2+σ3-2-1≔ma.s., which is the required statement. For the special case, if m<0, then it is concluded that (66)limt→∞ln⁡T∗tt=0,limt→∞ln⁡Vtt=0a.s., which means that limt→∞⁡T∗(t)=0 and limt→∞V(t)=0  a.s. In other words, the populations T∗ and V will tend to zero exponentially with probability one. This completes the proof.


## 5. Numerical Simulations

In Sections3 and 4, we obtained the stationary distribution and extinction of the virus under certain conditions, respectively. We now explore the cell and viral dynamics under different white noises. Here, by employing the Milstein’s higher order method in Higham [34], the discrete form of model (3) is (67)Tj+1=Tj+λ-dTj-kTjVjΔt+σ1TjΔtξj+σ122TjΔtξj2-Δt,Tj+1∗=Tj∗+kTjVj-δTj∗Δt+σ2Tj∗Δtηj+σ222Tj∗Δtηj2-Δt,Vj+1=Vj+NδTj∗-cVj-ikTjVjΔt+σ3VjΔtϕj+σ322VjΔtϕj2-Δt,i=0or1, where ξj, ηj, and ϕj, j=1,2,…,n, are the jth realization of three independent Gaussian random variables with distribution N(0,1), and the time increment Δt=0.015.In the following, for the sake of representativeness, we takei=1 for system (3) as an example, and we can carry out same simulations as follows when i=0. The parameters are chosen from the actual parameter values of previous references (see Table 1), and we calculate that the basic reproduction number R0,1=5.2609>1; from Section 2, we know that the chronic disease steady state is globally asymptotically stable. In order to examine how white noises affect model dynamic behavior, we choose three pairs of white noise set (σ1,σ2,σ3) equal to (0.008,0.3,1), (0.01,0.4,1.2), and (0.02,0.8,1.5), with the critical condition R0,1s equal to 3.0853, 2.6928, and 1.5107, respectively. For these parameter sets, Theorem 2 is satisfied and system (3) has a unique stationary distribution. By simulations, for each pair of white noise sets chosen above, Figures 1(a) and 2(a) demonstrate the solutions of the stochastic model (3) and the corresponding deterministic model (1) initiated at the initial value (T(0),T∗(0),V(0))=(1000,0.001,0.001), and Figures 1(b) and 2(b) show histograms of the approximate stationary distributions for two components (T and V) from 10000 simulation runs at time t=300. The figures indicate that the small magnitudes of random fluctuations can generate the irregular recurrence of populations T and V.Table 1
List of parameters.
Parameters
Description
Unit
Value
Source
λ
Target cells source term
μ
l
-
1 day−1
10
[5, 6, 9]
d
Death rate of healthy target cells
Day−1
0.007
[5, 6]
k
Viral infectivity rate
μl day−1
0.000024
[5, 6, 9]
δ
Death rate of infected target cells
Day−1
1
[5]
N
Burst term
Virions/cell
2000
[5, 6, 9]
c
Clearance rate of virus
Day−1
13
[5]Figure 1
(a) Simulations of healthy target cells for the stochastic model (3) and its corresponding deterministic model (1) with three different sets of white noise. (b) Probability density of the values of the path T(t) at t=300 for system (3) based on 10000 simulation runs. For parameters see Table 1 with (σ1,σ2,σ3) equal to (0.008,0.3,1), (0.01,0.4,1.2), and (0.02,0.8,1.5), respectively.
(a)
(b)Figure 2
(a) Simulations of virus for the stochastic model (3) and its corresponding deterministic model (1) with three different sets of white noise. (b) Probability density of the values of the path V(t) at t=300 for system (3) based on 10000 simulation runs. For parameters see Table 1 with (σ1,σ2,σ3) equal to (0.008,0.3,1), (0.01,0.4,1.2), and (0.02,0.8,1.5), respectively.
(a)
(b)Using the data values in Table1, we further choose two pairs of white noise set (σ1,σ2,σ3) equal to (0.03,1.2,2.5) and (0.06,1.8,3.5), with the critical condition R0,1s equal to 0.7850 and 0.2584, respectively. We do not give theoretical analysis in these cases, and now we take numerical simulation to discuss the effect of random fluctuations on model behavior. Figure 3 shows that, as the white noise set is increasing, the virus can be eradicated for stochastic model (3), while the virus is still persistent for its corresponding deterministic model (1). It means that the large magnitudes of random fluctuations could lead to the extinction of the virus.Figure 3
The paths ofV(t) for stochastic model (3) and the solutions of V(t) for its corresponding deterministic model (1) with (σ1,σ2,σ3) equal to (0.03,1.2,2.5) and (0.06,1.8,3.5), respectively.


## 6. Conclusions

We have studied a stochastic virus infection model with or without the virus loss term. Mathematically, we have given a systematic method to the analysis (stationary distribution and extinction) of such three-dimensional stochastic viral dynamic model. If the critical conditionR0,is (i=0 or 1) is greater than one, then system (3) has a unique stationary distribution. Furthermore, we obtain the sufficient conditions to ensure the extinction of the virus by employing the comparison theorem of stochastic differential equation and the law of large numbers. Numerical simulations demonstrate that, on one hand, healthy target cells and virions can be maintained within a certain range for the small magnitudes of white noises, which means that the virus is persistent; on the other hand, the virus could be cleared out for large magnitudes of white noises, which indicates the extinction of the virus.In fact, the theoretical analysis results of system (3) can also be extended to the following viral infection model with combination drug therapy:(68)dT=λ-dT-k1-nrtTVdt+σ1TdB1t,dT∗=k1-nrtTV-δT∗dt+σ2T∗dB2t,dV=N1-npδT∗-cV-ikTVdt+σ3VdB3t,i=0or1,where nrt and np are the drug efficacies of different types of drugs. Therefore, our model as well as the theoretical analysis results are extension of the literature [18].We have theoretically analyzed the existence of the stationary distribution and extinction for a stochastic HIV infection model with logistic T-cell and CTL immune response in literature [23]. Thus, with the work of this article, we may perform a similar study for our model in literature [5] with white noises: (69)dT=λ-dT+rT1-T+T∗Tmax-k1-nrtVTdt+σ1TdB1t,dT∗=k1-nrtVT-δT∗-qT∗Zdt+σ2T∗dB2t,dV=Nδ1-npT∗-c+ikTVdt+σ3VdB3t,dZ=pT∗Z-dZZdt+σ4ZdB4t,i=0or1, where Z denotes the cytotoxic T lymphocyte (CTL) immune response, q is the CTL activation rate, p is the CTL responsiveness, and dZ is the death rate of CTL. B4(t) is independent standard Brownian motion with B4(0)=0 and σ42>0 denote the intensity of the white noise. All the other parameters are the same as those in systems (1), (3), and (68). This kind of model will be our future work.



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*Source: 6027509-2017-10-22.xml*