Dynamics

The memory time ${\tau }_K$ is identified with the collision time ${\tau }_c$ by calculating the memory time using an Ornstein–Uhlenbeck model. The effective momentum-transfer cross-section is used to derive the relaxation time ${\tau }_r$ for various relativistic models with and without quantum effects and the radiation reaction. Models based on Møller scattering are also considered. The distinction between collision and relaxation times is established, as is the role they play in a non-Markovian relativistic generalized Langevin equation based on the Landau–Lifshitz equation. The memory time ${\tau }_K$ does not generally coincide with the relaxation time, but we show that in unphysical cases, they coincide. Relaxation times are calculated progressively by considering the differential cross-sections from various models ranging across the relativistic proposals of Rutherford; Huschilt and Baylis; Aguiar, Barroso, and Barone; and a modified Mott model. We also improve a Landau–Lifshitz model, but the effects due to the radiation reaction are negligible. Finally, the Møller model yields the best result, as it accounts for quantum effects in electron–electron collisions. Given the nondegenerate energy levels and fields considered here, a non-Markovian generalized Langevin equation based on the Landau–Lifshitz equation is derived.

Tires are of particular significance in vehicle studies, as they are the connection between the vehicle and the terrain. The tire model proposed in this study treats the tire as an elastic circular cylindrical shell on a nonlinear elastic foundation, with the goal of addressing the need for quick and accurate estimation of tire performance. The mechanics and structure of a tire are reflected in the model, creating strong connections among the physical properties of the tire and the model parameters. The forces and deformation equations are derived analytically with the energy method and solved with numerical approximations. The equations address nonlinear stiffness contributed by different parts of a tire, providing high accuracy in terms of spindle force prediction and pressure distribution without involving intense computational effort. The model is calibrated with experimental data to identify its parameters and is then used to simulate tire performance under other conditions. The simulated results are consistent with the experimental results, with $R^2$ values greater than 0.98.

We study a delayed reaction–diffusion susceptible–infectious–recovered (SIR) epidemic model with a Holling type-II incidence function. The model incorporates spatial diffusion and a discrete delay in the infectious input to the transmission term. We first formulate the model and discuss the existence and uniqueness of nonnegative solutions under suitable nonnegative initial histories. We then investigate the disease-free and endemic equilibria and derive the epidemic threshold by linearizing the infected equation at the disease-free equilibrium. For this normalised model the threshold quantity is given by $R_0$ the normalisation is such that the disease free susceptible level is equal to one. We prove the disease-free stability result for the strict subthreshold case $R_0<1$ and separate the critical case $R_0=1$ from this regime. The existence of a positive endemic equilibrium for $R_0>1$ is characterised. The stability of this equilibrium in the presence of a time delay requires that the characteristic equation and the global stability argument be consistent with the delay-dependent dynamics. The effects of diffusion, saturated incidence and delay are studied by numerical simulations. Transient oscillations are distinguished from sustained oscillations. The results presented here elucidate the assumptions and limitations of the proposed spatial epidemic model.