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Mathematical Biosciences and Engineering

2013, Volume 10, Issue 2: 483-498. doi: 10.3934/mbe.2013.10.483
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Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays

  • 1.
    College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082
  • 2.
    Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7
  • Received: 01 January 2012 Accepted: 29 June 2018 Published: 01 January 2013

  • MSC : Primary: 92D30; Secondary: 34C23.

  • We consider a mathematical model that describes the interactions ofthe HIV virus, CD4 cells and CTLs within host, which is amodification of some existing models by incorporating (i) twodistributed kernels reflecting the variance of time for virus toinvade into cells and the variance of time for invaded virions toreproduce within cells; (ii) a nonlinear incidence function $f$ forvirus infections, and (iii) a nonlinear removal rate function $h$for infected cells. By constructing Lyapunov functionals and subtleestimates of the derivatives of these Lyapunov functionals, we shownthat the model has the threshold dynamics: if the basicreproduction number (BRN) is less than or equal to one, then theinfection free equilibrium is globally asymptotically stable,meaning that HIV virus will be cleared; whereas if the BRN is largerthan one, then there exist an infected equilibrium which is globallyasymptotically stable, implying that the HIV-1 infection willpersist in the host and the viral concentration will approach apositive constant level. This together with thedependence/independence of the BRN on $f$ and $h$ reveals the effectof the adoption of these nonlinear functions.

    Citation: Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays[J]. Mathematical Biosciences and Engineering, 2013, 10(2): 483-498. doi: 10.3934/mbe.2013.10.483

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  • We consider a mathematical model that describes the interactions ofthe HIV virus, CD4 cells and CTLs within host, which is amodification of some existing models by incorporating (i) twodistributed kernels reflecting the variance of time for virus toinvade into cells and the variance of time for invaded virions toreproduce within cells; (ii) a nonlinear incidence function $f$ forvirus infections, and (iii) a nonlinear removal rate function $h$for infected cells. By constructing Lyapunov functionals and subtleestimates of the derivatives of these Lyapunov functionals, we shownthat the model has the threshold dynamics: if the basicreproduction number (BRN) is less than or equal to one, then theinfection free equilibrium is globally asymptotically stable,meaning that HIV virus will be cleared; whereas if the BRN is largerthan one, then there exist an infected equilibrium which is globallyasymptotically stable, implying that the HIV-1 infection willpersist in the host and the viral concentration will approach apositive constant level. This together with thedependence/independence of the BRN on $f$ and $h$ reveals the effectof the adoption of these nonlinear functions.


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