Global dynamics of an age-structured within-host viral infection model with cell-to-cell transmission and general humoral immunity response

Ran Zhang; Shengqiang Liu; Ran Zhang; Shengqiang Liu

doi:10.3934/mbe.2020075

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1450-1478. doi: 10.3934/mbe.2020075

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Global dynamics of an age-structured within-host viral infection model with cell-to-cell transmission and general humoral immunity response

Ran Zhang ¹,
Shengqiang Liu ^{2
,
,}

1.
School of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2.
School of Mathematical Sciences, Tiangong University, Tianjin 300387, China

Received: 11 October 2019 Accepted: 22 November 2019 Published: 28 November 2019

In this paper, an age-structured within-host viral infection model with cell-to-cell transmission and general humoral immune response is investigated. We give a rigorous mathematical analysis on some necessary technical materials, including the relative compactness and persistence of the solution semiflow, and existence of a global attractor. By subtle construction and estimates of a Lyapunov functional, we show that the global dynamics is determined by two sharp thresholds, namely, basic reproduction number $\Re_0$ and immune-response reproduction number $\Re_1$. When $\Re_0 < 1$, the virus-free steady state is globally asymptotically stable, which means that the viruses are cleared and immune-response is not active; when $\Re_1 < 1 < \Re_0$, the immune-inactivated infection steady state exists and is globally asymptotically stable; and when $\Re_1>1$, which implies that $\Re_0>1$, the immune-activated infection steady state exists and is globally asymptotically stable. Numerical simulations are given to support our theoretical results.
- age structure,
- viral infection model,
- immune response,
- stability,
- Lyapunov functional
Citation: Ran Zhang, Shengqiang Liu. Global dynamics of an age-structured within-host viral infection model with cell-to-cell transmission and general humoral immunity response[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1450-1478. doi: 10.3934/mbe.2020075

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Abstract

In this paper, an age-structured within-host viral infection model with cell-to-cell transmission and general humoral immune response is investigated. We give a rigorous mathematical analysis on some necessary technical materials, including the relative compactness and persistence of the solution semiflow, and existence of a global attractor. By subtle construction and estimates of a Lyapunov functional, we show that the global dynamics is determined by two sharp thresholds, namely, basic reproduction number $\Re_0$ and immune-response reproduction number $\Re_1$. When $\Re_0 < 1$, the virus-free steady state is globally asymptotically stable, which means that the viruses are cleared and immune-response is not active; when $\Re_1 < 1 < \Re_0$, the immune-inactivated infection steady state exists and is globally asymptotically stable; and when $\Re_1>1$, which implies that $\Re_0>1$, the immune-activated infection steady state exists and is globally asymptotically stable. Numerical simulations are given to support our theoretical results.

References

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