Stochastic dynamics in a time-delayed model for autoimmunity

Time-delayed and stochastic effects in a predator-prey model with ratio dependence and Holling type III functional response

In this article, we derive and analyze a novel predator-prey model with account for maturation delay in predators, ratio dependence, and Holling type III functional response. The analysis of the system's steady states reveals conditions on predation rate, predator growth rate, and maturation time that can result in a prey-only equilibrium or facilitate simultaneous survival of prey and predators in the form of a stable coexistence steady state, or sustain periodic oscillations around this state. Demographic stochasticity in the model is explored by means of deriving a delayed chemical master equation. Using system size expansion, we study the structure of stochastic oscillations around the deterministically stable coexistence state by analyzing the dependence of variance and coherence of stochastic oscillations on system parameters. Numerical simulations of the stochastic model are performed to illustrate stochastic amplification, where individual stochastic realizations can exhibit sustained oscillations in the case, where deterministically the system approaches a stable steady state. These results provide a framework for studying realistic predator-prey systems with Holling type III functional response in the presence of stochasticity, where an important role is played by non-negligible predator maturation delay.

Quantifying the Role of Stochasticity in the Development of Autoimmune Disease

In this paper, we propose and analyse a mathematical model for the onset and development of autoimmune disease, with particular attention to stochastic effects in the dynamics. Stability analysis yields parameter regions associated with normal cell homeostasis, or sustained periodic oscillations. Variance of these oscillations and the effects of stochastic amplification are also explored. Theoretical results are complemented by experiments, in which experimental autoimmune uveoretinitis (EAU) was induced in B10.RIII and C57BL/6 mice. For both cases, we discuss peculiarities of disease development, the levels of variation in T cell populations in a population of genetically identical organisms, as well as a comparison with model outputs.