Department of Mathematics, University of York, York YO10 5DD, UK.
Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, UK.
Math Biosci. 2020 Apr;322:108323. doi: 10.1016/j.mbs.2020.108323. Epub 2020 Feb 22.
In this paper we study interactions between stochasticity and time delays in the dynamics of immune response to viral infections, with particular interest in the onset and development of autoimmune response. Starting with a deterministic time-delayed model of immune response to infection, which includes cytokines and T cells with different activation thresholds, we derive an exact delayed chemical master equation for the probability density. We use system size expansion and linear noise approximation to explore how variance and coherence of stochastic oscillations depend on parameters, and to show that stochastic oscillations become more regular when regulatory T cells become more effective at clearing autoreactive T cells. Reformulating the model as an Itô stochastic delay differential equation, we perform numerical simulations to illustrate the dynamics of the model and associated probability distributions in different parameter regimes. The results suggest that even in cases where the deterministic model has stable steady states, in individual stochastic realisations, the model can exhibit sustained stochastic oscillations, whose variance increases as one gets closer to the deterministic stability boundary. Furthermore, in the regime of bi-stability, whereas deterministically the system would approach one of the steady states (or periodic solutions) depending on the initial conditions, due to the presence of stochasticity, it is now possible for the system to reach both of those dynamical states with certain probability. Biological significance of this result lies in highlighting the fact that since normally in a laboratory or clinical setting one would observe a single individual realisation of the course of the disease, even for all parameters characterising the immune system and the strength of infection being the same, there is a proportion of cases where a spontaneous recovery can be observed, and similarly, where a disease can develop in a situation that otherwise would result in a normal disease clearance.
本文研究了病毒感染免疫反应动力学中随机性和时滞的相互作用，特别关注自身免疫反应的发生和发展。我们从感染免疫反应的确定性时滞模型开始，该模型包括具有不同激活阈值的细胞因子和 T 细胞，为概率密度推导出了精确的时滞化学主方程。我们使用系统大小展开和线性噪声近似来研究随机振荡的方差和相干性如何随参数变化，并表明当调节性 T 细胞更有效地清除自身反应性 T 细胞时，随机振荡变得更加规则。将模型重新表述为 Ito 随机时滞微分方程，我们进行数值模拟以说明模型的动力学及其在不同参数区域的相关概率分布。结果表明，即使在确定性模型具有稳定平衡点的情况下，在单个随机实现中，模型也可以表现出持续的随机振荡，其方差随着接近确定性稳定性边界而增加。此外，在双稳定性区域，尽管系统在确定性情况下会根据初始条件接近其中一个稳定状态（或周期解），但由于存在随机性，现在系统有可能以一定的概率达到这两个动态状态。这一结果的生物学意义在于强调这样一个事实，即在实验室或临床环境中，通常会观察到疾病过程的单个个体实现，即使免疫系统和感染强度的所有参数都相同，也有一部分病例可以观察到自发恢复，同样，在某些情况下，疾病也可以在原本会导致正常疾病清除的情况下发展。
