Mathematical modelling of oscillating patterns for chronic autoimmune diseases

Many autoimmune diseases are chronic in nature, so that in general patients experience periods of recurrence and remission of the symptoms characterizing their specific autoimmune ailment. In order to describe this very important feature of autoimmunity, we construct a mathematical model of kinetic type describing the immune system cellular interactions in the context of autoimmunity exhibiting recurrent dynamics. The model equations constitute a non-linear system of integro-differential equations with quadratic terms that describe the interactions between self-antigen presenting cells, self-reactive T cells and immunosuppressive cells. We consider a constant input of self-antigen presenting cells, due to external environmental factors that are believed to trigger autoimmunity in people with predisposition for this condition. We also consider the natural death of all cell populations involved in our model, caused by their interaction with cells of the host environment. We derive the macroscopic analogue and show positivity and well-posedness of the solution, and then we study the equilibria of the corresponding dynamical system and their stability properties. By applying dynamical system theory, we prove that steady oscillations may arise due to the occurrence of a Hopf bifurcation. We perform some numerical simulations for our model, and we observe a recurrent pattern in the solutions of both the kinetic description and its macroscopic analogue, which leads us to conclude that this model is able to capture the chronic behaviour of many autoimmune diseases.

On a Method in Dynamic Elasticity Problems for Heterogeneous Wedge-Shaped Medium

The method of analysis of steady oscillations arising in the piecewise homogeneous wedge-shaped medium composed by two homogeneous elastic wedges with different mechanical and geometric characteristics is presented. Method is based on the distributions’ integral transform technique and allows reconstructing the wave field in the whole medium by displacement oscillations given in the domain on the boundary of the medium. The problem in question is reduced to a boundary integral equation (BIA). Solvability problems of the BIA are examined and the structure of its solution is established.

On modeling bodies with delaminating coatings taking into account the fields of prestresses

The paper presents a model of steady-state oscillations of an inhomogeneous body with a prestressed exfoliating coating based on a general linearized statement of the problem of the motion of a prestressed-strained elastic body. On its basis, the statement of the problem of oscillations of an inhomogeneous strip consisting of a substrate and a prestressed coating is formulated, between which there is a delamination in a certain region. Steady oscillations are caused by a load applied to the upper boundary of the coating. To calculate the oscillations of the two-dimensional structure under consideration, the Fourier transform in the longitudinal coordinate was used and the original problem was reduced to solving a number of auxiliary boundary value problems with respect to transformants of the desired functions. From the conditions that the stress functions vanish (the cover is modeled as a mathematical section) of the substrate and the coating, the operator relations are constructed in the area of delamination to calculate the opening functions. The kernels of these operator relations are singular and are integrals over an infinite interval. A study was made of the behavior of their integrands at infinity, on the basis of which special approaches were used to calculate the kernels. As a result of solving the obtained hypersingular equations with difference kernels, for which the collocation method is used, the originals of the disclosure functions are constructed. Using a similar approach for inverting the Fourier transformations, we constructed relations to calculate the originals of the displacement functions at the upper boundary of the coverage. Based on the computational experiments, an analysis is made of the influence of the initial geometric and mechanical parameters of the substrate and coating on the values of the disclosure functions in the delamination region and the displacement functions at the upper boundary of the layer. The influence of the prestress level on the amplitude-frequency characteristics (AFC) was also investigated. It was found that the most significant effect on the frequency response is in the vicinity of the frequencies of the thick resonances. Based on the information on the displacement fields, it is possible to construct schemes for identifying delamination characteristics.

Couple stress fluid flow due to slow steady oscillations of a permeable sphere

The study of oscillating flow of a Couple Stress fluid past a permeable sphere is considered. Analytical solution for the flow field in terms of stream function is obtained using modified Bessel functions. The formula for Drag acting on the sphere due external flow is evaluated. Pressure field for the flow region past and inside the sphere is obtained. Effects of physical parameters like couple stress parameter, permeability, frequency and geometric parameters on the drag due to internal and external flows are represented graphically. It is observed that the drag for viscous fluid flow will be less than the case of couple-stress fluid flow and hence couple stress fluids offer resistance for flow.

Synthesis of model predictive controller for an identified model of MIMO process

Model Predictive Controller (MPC) technology has been researched and developed to meet varied demands of need to control industrial power plants and petroleum refineries. This development has paved the way for the MPC technology too many other fields like automotive, aerospace, food processing industries in this paper, primary importance has been paid to the development of a MPC for an identified model of Multiple Input and Multiple Output process. In this paper, a Four Tank System has been considered for generation of input-output data. This data i.e. generated input output data is used for the estimation of two polynomial model, name ARX model (Autoregressive exogenous) model and OE (Output Error) model. With each of model output generated, the Fit-Rates of models are compared to find out most efficient model. The model equations are now considered as plant for developing a Model Predictive Controller (MPC). Two sets of results are obtained after the development of MPC and tested. One is without noise and one is with noise. Both sets of results were a success as the output signals traces step input signals after some steady oscillations in real time with in a very short period of time which indicated a good response time. The MPC developed can be applied to any polynomial model with a good Fit-Rate, it predicts and control the process variables automatically.