In this paper the model of infection diseases by Marchuk is considered. Mathematical questions which are important in its study are discussed. Among them there are stability of stationary points, construction of the Cauchy matrices of linearized models, estimates of solutions. The novelty we propose is in a distributed feedback control which affects the antibody concentration. We use this control in the form of an integral term and come to the analysis of nonlinear integro-differential systems. New methods for the study of stability of linearized integro–differential systems describing the model of infection diseases are proposed. Explicit conditions of the exponential stability of the stationary points characterizing the state of the healthy body are obtained. The method of the paper is based on the symmetry properties of the Cauchy matrices which allow us their construction.

A well-known model of infectious diseases, described by a nonlinear system of delay differential equations, is investigated under the influence of stochastic perturbations. Using the general method of Lyapunov functional construction combined with the linear matrix inequality (LMI) approach, we derive sufficient conditions for the stability of the equilibria of the considered system. Numerical simulations illustrating the system’s behavior under stochastic perturbations are provided to support the thoretical findings. The proposed method for stability analysis is broadly applicable to other systems of nonlinear stochastic differential equations across various fields.

Taking the Singularly Perturbed System (SPS) as a model of ODE system separation into fast and slow subsystems by an arbitrarily small parameter, we state and prove a theorem on the decomposition of an Ordinary Differential Equations (ODE) system without the aforementioned arbitrarily small parameter. In accordance with the proven theorem, we implemented an algorithm to decompose an ODE system into fast and slow subsystems by coordinate transformation. A similar algorithm is called the Singular Perturbed Vector Field (SPVF) algorithm; however, it is not justified by any stated theorem. Since we have not found any theorem to propose a similar ODE decomposition in the literature, we have tried to fill the gap with our theorem and algorithm explanations through examples. Finally, we propose our concept on Marchuk’s infectious diseases model, which allows a different analysis of the original Marchuk’s ODE system with delay.

We coin the term “network goodness” for a value we define for a network embedded in a given environment as a metric that describes the suitability of that network for meeting a demand. Three formulas are proposed to calculate the metric from three variable values. The first variable considers parts of the environment gravitated by the network. For these parts of the environment, we define a value that measures user costs refusing them the use of the network. Last but not least, the network maintenance costs are considered. The results are obtained after focusing on infrastructure and transport networks, but can be used for other types of networks as well.

We analyze parameter identifiability in a Marchuk-type immune-response model using longitudinal whole-blood transcriptomic signatures from the influenza challenge. Latent states are extracted from curated gene signatures derived from nine symptomatic and eight asymptomatic subjects. The governing delay differential equations are cast in a linear-in-parameters form; derivatives are estimated by smoothing splines, coefficients are fit by ridge regression, and the delay $\tau$ is selected by grid search. We find that the parameters governing viral and innate dynamics are consistently identifiable, with low relative error, and are highly determined, whereas adaptive-immunity and tissue-damage parameters are poorly constrained by transcriptomics alone. Introducing a small additive background term and tissue dependence markedly reduces residual variance and stabilizes estimates. Symptomatic patients exhibit a characteristic regulatory delay near 21 h. These results show that aggregated transcriptomic time series can reliably identify some subsystems of classical immune models, but that adaptive immunity and damage dynamics require explicit structural extensions or additional data modalities. The study provides a practical identification pipeline and concrete guidance on model extensions needed for transcriptomic-driven mechanistic inference.