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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-1β 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 R0>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 R0<1, that is, it can induce the extinction of virus, regardless of whether or not the parameters are spatially dependent. [ABSTRACT FROM AUTHOR]