Bulletin of Mathematical Biology

, Volume 80, Issue 3, pp 540–582 | Cite as

Caspase-1-Mediated Pyroptosis of the Predominance for Driving CD4\(^{+}\) T Cells Death: A Nonlocal Spatial Mathematical Model

  • Wei Wang
  • Tongqian Zhang
Original Article


Caspase-1-mediated pyroptosis is the predominance for driving CD4\(^{+}\) T cells death. Dying infected CD4\(^{+}\) T cells can release inflammatory signals which attract more uninfected CD4\(^{+}\) T cells to die. This paper is devoted to developing a diffusive mathematical model which can make useful contributions to understanding caspase-1-mediated pyroptosis by inflammatory cytokines IL-1\(\beta \) released from infected cells in the within-host environment. The well-posedness of solutions, basic reproduction number, threshold dynamics are investigated for spatially heterogeneous infection. Travelling wave solutions for spatially homogeneous infection are studied. Numerical computations reveal that the spatially heterogeneous infection can make \(\mathscr {R}_0>1\), that is, it can induce the persistence of virus compared to the spatially homogeneous infection. We also find that the random movements of virus have no effect on basic reproduction number for the spatially homogeneous model, while it may result in less infection risk for the spatially heterogeneous model, under some suitable parameters. Further, the death of infected CD4\(^{+}\) cells which are caused by pyroptosis can make \(\mathscr {R}_0<1\), that is, it can induce the extinction of virus, regardless of whether or not the parameters are spatially dependent.


Caspase-1-mediated pyroptosis Inflammatory cytokines IL-1\(\beta \) Basic reproduction number Threshold dynamics Travelling wave solutions 

Mathematics Subject Classification

92D40 34K18 34D20 



This work is supported by the National Natural Science Foundation of China (No. 11371230), Shandong Provincial Natural Science Foundation, China (No. ZR2015AQ001), a Project for Higher Educational Science and Technology Program of Shandong Province of China (No. J13LI05), SDUST Research Fund (2014TDJH102), and Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province of China.


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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of Applied Mathematics, School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingPeople’s Republic of China
  2. 2.Department of Mathematics, College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoPeople’s Republic of China
  3. 3.State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and TechnologyShandong University of Science and TechnologyQingdaoPeople’s Republic of China

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