Global Asymptotic Stability and Nonlinear Analysis of the Model of the Square Immunopixels Array Based on Delay Lattice Differential Equations

Abstract

Biosensors and immunosensors show an increasing attractiveness when developing current cheap and fast monitoring and detecting devices. In this work, a model of immunosensor in a class of delayed lattice differential equations is offered and studied. The spatial operator describes symmetric diffusion processes of antigenes between pixels. The main results are devoted to the qualitative research of the model. The conditions of global asymptotic stability, which are constructed with the help of Lyapunov functionals, determine a lower estimate of the time of immune response. Nonlinear analysis of the model is performed with help of a series of numerical characteristics including autocorrelation function, mutual information, embedding, and correlation dimensions, sample entropy, the largest Lyapunov exponents. We consider the influence of both symmetric and unsymmetric diffusion of antigens between pixels on the qualitative behavior of the system. The outcomes are verified with the help of numerical simulation in cases of 4 × 4 - and 16 × 16 - arrays of immunopixels.

On qualitative analysis of the nonstationary delayed model of coexistence of two-strain virus: Stability, bifurcation, and transition to chaos

“…Complete research of qualitative behavior of the model can be executed with the help of investigation of the corresponding time series based on characteristics of nonlinear dynamics. Here we will follow to the general computational approach which was developed and applied earlier in the work [17] .…”

Age-structured delayed SIPCV epidemic model of HPV and cervical cancer cells dynamics I. Numerical method

“…Using the method of characteristics [3], [4], [5], [6], [8], [9], [11], [24], [53], [55], [58] and method of steps from the theory of delayed differential equations [12], [28], [46], [57], [59], we obtain an exact solution of the SIPCV epidemic model. This solution is given in form of the recurrent formulae (like in works [3], [4], [8], [55]) in which the densities of all subpopulations are defined through the integrals from solution taken at previous instance of time.…”