Stochastic dynamics in a time-delayed model for autoimmunity

Abstract

Article (Accepted Version) http://sro.sussex.ac.uk Fatehi, Farzad, Kyrychko, Yuliya N and Blyuss, Konstantin B (2020) Stochastic dynamics in a time-delayed model for autoimmunity. Mathematical Biosciences, 322. a108323.

Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response

“…It would be interesting and important to investigate how (in)stability of extinction steady state is affected by stochasticity, and how it impacts the regimes of bi-stability between extinction and other equilibria. This could be achieved through numerical exploration of basins of attraction of different steady states in the stochastic model in a manner similar to how it was done by Fatehi et al [47,48] in the context of modelling autoimmunity arising from immune response to a viral infection.…”

Quantifying the Role of Stochasticity in the Development of Autoimmune Disease

“…However, earlier work of Iwami et al [10,11] has shown that, while this behaviour can be observed for both types of growth functions in the absence of infection or immune reaction, their specific functional form does have a significant effect on the overall dynamics of autoimmune disease. Thus, we have chosen to use a logistic form for the growth function of healthy organ cells, in agreement with Iwami et al [10] and our earlier work on autoimmunity [34][35][36][37][38][39][40].…”

“…Blyuss and Nicholson [34,35] used a TAT-based model to investigate autoimmunity arising through a mechanism of molecular mimicry from immune response to a viral infection. To capture a dynamical regime where autoimmunity arises as a by-product of viral infection but after that initial infection has already been cleared by the immune system, Fatehi et al [36][37][38][39][40] developed this model further by including cytokines mediating T cell proliferation, as well as time delays associated with various aspects of the immune response.…”

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

“…at order Ω 0 one obtains a delayed Fokker-Planck equation that describes stochastic oscillations around the deterministic trajectory (see Appendix A). Following the methodology of Galla 60 (see also [61][62][63][64] ), we can use this delayed Fokker-Planck equation to obtain the following system of Langevin equations describing the dynamics of fluctuations around a deterministic steady state (u * , v * ) of the model ( 2)…”

“…while for stochastic systems without time delay one could use a Lyapunov equation to obtain the covariance matrix 61,66,67 . The level of fluctuations around the dominant spectral frequency for each of the two populations X u (t) and X v (t) around their steady-state values X * u and X * v can be quantified using their respective mean-square variances…”