Global dynamics of an immunosuppressive infection model with stage structure

Hongying Shu; Wanxiao Xu; Zenghui Hao; Hongying Shu; Wanxiao Xu; Zenghui Hao

doi:10.3934/mbe.2020111

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 3: 2082-2102. doi: 10.3934/mbe.2020111

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Global dynamics of an immunosuppressive infection model with stage structure

1.
School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China
2.
Department of Mathematics, Tongji University, Shanghai 200092, China

Received: 29 September 2019 Accepted: 23 December 2019 Published: 03 January 2020

In this paper, we propose an immunosuppressive infection model incorporating natural mortality of immune cells during the time lag needed for the expansion of immune cells. Starting from a stage structure model for the immune cells with various ages, we use the method of characteristic lines to derive a delay differential equation for the population of mature immune cells. Then, we use Lyapunov functional techniques to obtain the global dynamics of the model system. Specifically, we show that the virus dominant equilibrium is globally asymptotically stable when the delay is large. Next, we conduct local and global Hopf bifurcation analysis for the proposed model via Hopf bifurcation theory of delay differential equations. We choose the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations of periodic solutions from the immune control equilibrium. We also prove that the model has only a finite number of Hopf bifurcation values, and the periodic solutions with specific oscillation frequencies occur only in bounded delay intervals. Under some technical conditions, we show that two global Hopf branches bifurcated from different Hopf bifurcation values may connect to each other and thus be bounded. However, unlike the global Hopf bifurcation results in the existing literature, the Hopf branches for our model system are not necessarily bounded, though the delay components are always bounded. Numerical simulation suggests that bounded and unbounded Hopf branches may co-exist in the bifurcation diagram. Moreover, we observe a new interesting phenomenon that a global Hopf branch may have uniformly bounded periodic solutions, bounded delays, but unbounded periods.
- delayed antiviral immune response,
- Stage structure,
- Hopf bifurcation,
- Periodic solutions,
- Hopf branch
Citation: Hongying Shu, Wanxiao Xu, Zenghui Hao. Global dynamics of an immunosuppressive infection model with stage structure[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2082-2102. doi: 10.3934/mbe.2020111

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Abstract

In this paper, we propose an immunosuppressive infection model incorporating natural mortality of immune cells during the time lag needed for the expansion of immune cells. Starting from a stage structure model for the immune cells with various ages, we use the method of characteristic lines to derive a delay differential equation for the population of mature immune cells. Then, we use Lyapunov functional techniques to obtain the global dynamics of the model system. Specifically, we show that the virus dominant equilibrium is globally asymptotically stable when the delay is large. Next, we conduct local and global Hopf bifurcation analysis for the proposed model via Hopf bifurcation theory of delay differential equations. We choose the delay as a bifurcation parameter and examine the onset and termination of Hopf bifurcations of periodic solutions from the immune control equilibrium. We also prove that the model has only a finite number of Hopf bifurcation values, and the periodic solutions with specific oscillation frequencies occur only in bounded delay intervals. Under some technical conditions, we show that two global Hopf branches bifurcated from different Hopf bifurcation values may connect to each other and thus be bounded. However, unlike the global Hopf bifurcation results in the existing literature, the Hopf branches for our model system are not necessarily bounded, though the delay components are always bounded. Numerical simulation suggests that bounded and unbounded Hopf branches may co-exist in the bifurcation diagram. Moreover, we observe a new interesting phenomenon that a global Hopf branch may have uniformly bounded periodic solutions, bounded delays, but unbounded periods.

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