• Chinese Physics B
  • Vol. 29, Issue 9, (2020)
Dong-Xi Li1,† and Ni Zhang2
Author Affiliations
  • 1College of Data Science, Taiyuan University of Technology, Taiyuan 030024, China
  • 2College of Mathematics, Taiyuan University of Technology, Taiyuan 03004, China
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    DOI: 10.1088/1674-1056/ab8c3f Cite this Article
    Dong-Xi Li, Ni Zhang. Dynamical analysis for hybrid virus infection system in switching environment[J]. Chinese Physics B, 2020, 29(9): Copy Citation Text show less

    Abstract

    We investigate the dynamical behavior of hybrid virus infection systems with nonlytic immune response in switching environment, which is modeled as a stochastic process of telegraph noise and represented as a multi-state Markov chains. Firstly, The existence of unique positive solution and boundedness of the new hybrid system is proved. Furthermore, the sufficient conditions for extinction and persistence of virus are established. Finally, stochastic simulations are performed to test and demonstrate the conclusions. As a consequence, our work suggests that stochastic switching environment plays a crucial role in the process of virus prevention and treatment.
    {x˙(t)=λδx(t)βx(t)y(t)1+qz(t),y˙(t)=βx(t)y(t)1+qz(t)ay(t)py(t)z(t),z˙(t)=cy(t)bz(t).(1)

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    P{r(t+Δt)=j|r(t)=i}={νijΔt+o(Δt),ifij,1+νiiΔt+o(Δt),ifi=j,(2)

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    π1=ν21ν12+ν21andπ2=ν12ν12+ν21.()

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    ν^=maxkSν(k),νˇ=minkSν(k).()

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    {x˙(t)=λr(t)δr(t)x(t)βr(t)x(t)y(t)1+qr(t)z(t),y˙(t)=βr(t)x(t)y(t)1+qr(t)z(t)ar(t)y(t)pr(t)y(t)z(t),z˙(t)=cr(t)y(t)br(t)z(t),(3)

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    x˙(t)=λr(t)δr(t)x(t)βr(t)x(t)y(t)1+qr(t)z(t)<λ^δˇx(t).(4)

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    x(t)<λ^δˇ+1,foralllarget,sayt>t0.()

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    x˙(t)+y˙(t)=λr(t)δr(t)x(t)ar(t)y(t)pr(t)y(t)z(t)<λ^δˇx(t).()

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    x(t)+y(t)C+λ^δˇ+1,t>t0,()

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    x(t)+y(t)<C+λ^δˇ+1,forallt>t1.()

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    z˙(t)<c^(C+λ^δˇ+1)bˇz,forlarget()

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    E1={x¯,y¯,z¯}={cλ(1+qz¯)cδ+(bβ+cδq)z¯,bz¯c,pcδ+abβ+acδq2(bpβ+cδpq)+(pcδ+abβ+acδq)24p(bβ+cδq)(acδcλβ)2(bpβ+cδpq)},(5)

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    αr(t)=βr(t)λr(t)δr(t)ar(t).()

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    R0=(π1β1+π2β2)(π1λ1+π2λ2)(π1a1+π2a2)(π1δ1+π2δ2)<1,()

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    liminftx(t)π1λ1+π2λ2π1δ1+π2δ2a.s.(6)

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    limsuptx(t)π1λ1+π2λ2π1δ1+π2δ2a.s.(7)

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    limty(t)=0a.s.(8)

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    limtz(t)=0a.s.(9)

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    Ω1={ωΩ:liminftx(t)>π1λ1+π2λ2π1δ1+π2δ2+ε}.(10)

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    limt1t(λr(s)δr(s)[π1λ1+π2λ2π1δ1+π2δ2+ε])ds=π1(λ1δ1[π1λ1+π2λ2π1δ1+π2δ2+ε])+π2(λ2δ2[π1λ1+π2λ2π1δ1+π2δ2+ε])=(π1δ1+π2δ2)ε.(11)

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    dx(t)=[λr(t)δr(t)x(t)+βr(t)x(t)y(t)qr(t)z(t)+1]dt(λr(t)δr(t)x(t))dt.(12)

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    x(t)x(0)+0T(λr(s)δr(s)x(s))ds+Tt(λr(s)δr(s)[π1λ1+π2λ2π1δ1+π2δ2+ε])ds()

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    limsuptx(t)t(π1δ1+π2δ2)ε,()

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    limtx(t)=0.()

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    x(t)x(0)t=1t0t(λr(s)δr(s)x(s)βr(s)x(s)y(s)qr(s)z(s)+1)ds1t0t(λr(s)δr(s)x(s))ds.()

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    1t0tδr(s)x(s)ds1t0tλr(s)dsx(t)x(0)t.(13)

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    (π1δ1+π2δ2)limt1t0tx(s)dsπ1λ1+π2λ2.()

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    limt1t0tx(s)dsπ1λ1+π2λ2π1δ1+π2δ2+ε.()

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    dlny(t)=[βr(t)x(t)qr(t)z(t)+1ar(t)pr(t)z(t)]dt.()

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    lny(t)lny(0)t=1t0t(ar(s)+βr(s)x(s)qr(s)z(s)+1pr(s)z(s))ds1t0t(βr(s)x(s)ar(s))ds.(14)

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    limsupt1tln(y(t))[π1β1+π2β2]π1λ1+π2λ2π1δ1+π2δ2(π1a1+π2a2).()

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    (π1β1+π2β2)(π1λ1+π2λ2)<(π1a1+π2a2)(π1δ1+π2δ2),()

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    limty(t)=0.()

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    Ω3={ωΩ:limsuptx(t)<π1λ1+π2λ2π1δ1+π2δ2ε}.(15)

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    limt1t0t(λr(s)δr(s)[π1λ1+π2λ2π1δ1+π2δ2ε])ds=π1(λ1δ1[π1λ1+π2λ2π1δ1+π2δ2ε])+π2(λ2δ2[π1λ1+π2λ2π1δ1+π2δ2ε])=(π1δ1+π2δ2)ε.()

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    x(t)x(0)=0t(λr(s)δr(s)x(s))ds0tβr(s)x(s)y(s)qr(s)z(s)+1ds0T(λr(s)δr(s)x(s))ds+Tt(λr(s)δr(s)[π1λ1+π2λ2π1δ1+π2δ2])ds0tβr(s)x(s)y(s)qr(s)z(s)+1ds()

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    limsuptx(t)t(π1δ1+π2δ2)ε,()

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    limtx(t)=.()

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    R0=(π1β1+π2β2)(π1λ1+π2λ2)(π1a1+π2a2)(π1δ1+π2δ2)>1,()

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    y(t)y(0)t=1t0t[βr(s)x(s)qr(s)z(s)+1ar(s)y(s)pr(s)y(s)z(s)]ds.(16)

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    x(t)x(0)t+y(t)y(0)t=1t0t(λr(s)δr(s)x(s)ar(s)y(s)pr(s)y(s)z(s))ds.(17)

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    1t0tλr(s)ds=1t0t(δr(s)x(s))dsx(t)x(0)ty(t)y(0)t1t0t(ar(s)y(s))ds1t0t(pr(s)y(s)z(s))ds.()

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    limt1t0tx(s)ds=π1λ1+π2λ2π1δ1+π2δ2π1a1+π2a2π1δ1+π2δ2limt1t0ty(s)ds1π1δ1+π2δ2limt1t0t(pr(s)y(s)z(s))dsπ1λ1+π2λ2π1δ1+π2δ2π1a1+π2a2π1δ1+π2δ2limt1t0ty(s)ds.(18)

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    1t0t(βr(s)x(s))dslny(t)lny(0)t+1t0tar(s)ds.()

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    1t0tx(s)ds1π1β1+π2β2lny(t)lny(0)t+π1a1+π2a2π1β1+π2β2.(19)

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    π1a1+π2a2π1β1+π2β2π1λ1+π2λ2π1δ1+π2δ2π1a1+π2a2π1δ1+π2δ2limt1t0ty(s)ds.(20)

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    liminft1t0ty(s)ds((π1β1+π2β2)(π1λ1+π2λ2)(π1δ1+π2δ2)(π1a1+π2a2)(π1β1+π2β2)(π1δ1+π2δ2))(π1β1+π2β2π1a1+π2a2).()

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    (π1β1+π2β2)(π1λ1+π2λ2)>(π1a1+π2a2)(π1δ1+π2δ2),()

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    liminft1t0ty(s)ds>0a.s.()

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    xk+1=xk+(λr(t)δr(t)x(t)βr(t)x(t)y(t)1+qr(t)z(t))Δt,yk+1=yk+(βr(t)x(t)y(t)1+qr(t)z(t)ar(t)y(t)pr(t)y(t)z(t))Δt,zk+1=zk+(cr(t)y(t)br(t)z(t))Δt.()

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    E={x,y,z}={π1c1λ1(1+q1z¯1)+π2c2λ2(1+q2z¯2)π1c1δ1+(b1β1+c1δ1q1)z¯1+π2c2δ2+(b2β2+c2δ2q2)z¯2,π1b1z¯1+π2b2z¯2π1c1+π2c2,π1((p1c1δ1+a1b1β1+a1c1δ1q1))2π1(b1p1β1+c1δ1p1q1)+2π2(b2p2β2+c2δ2p2q2)+π1((p1c1δ1+a1b1β1+a1c1δ1q)24p1(b1β1+c1δ1q1)(a1c1δ1c1λ1β1))2π1(b1p1β1+c1δ1p1q1)+2π2(b2p2β2+c2δ2p2q2)+π2((p2c2δ2+a2b2β2+a2c2δ2q2))2π1(b1p1β1+c1δ1p1q1)+2π2(b2p2β2+c2δ2p2q2)+π2((p2c2δ2+a2b2β2+a2c2δ2q)24p2(b2β2+c2δ2q2)(a2c2δ2c2λ2β2))2π1(b1p1β1+c1δ1p1q1)+2π2(b2p2β2+c2δ2p2q2)}.()

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    Dong-Xi Li, Ni Zhang. Dynamical analysis for hybrid virus infection system in switching environment[J]. Chinese Physics B, 2020, 29(9):
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