Using Differential Equations with Time Delay on a Hexagonal Lattice for Modeling Immunosensors
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USING DIFFERENTIAL EQUATIONS WITH TIME DELAY ON A HEXAGONAL LATTICE FOR MODELING IMMUNOSENSORS V. Martsenyuk,1 A. Sverstiuk,2† and I. S. Gvozdetska2‡
UDC 602.1:519.85:53.082.9:616-07
Abstract. A model of an immunosensor is proposed based on a system of differential equations with time delay on a hexagonal lattice. The presented main result consists of conditions of local asymptotic stability of an endemic state. To obtain this result, the method of Lyapunov functionals is used. It combines the general approach to constructing Lyapunov functionals for predator-prey models and differential equations with time delay on a hexagonal lattice. A numerical example shows the influence of time delays on stability, namely, the transition from a stable focus to a limit cycle through a Hopf bifurcation occurs. Keywords: biosensor, immunosensor, differential equation on a hexagonal lattice, differential equation with time delay, asymptotic stability, Lyapunov functional. INTRODUCTION In the past two decades, interest in research in the field of biosensors increases. It is connected, first of all, with their specificity and high sensitivity that is reached due to the use of biological molecules and systems. Biosensors are analytical instruments containing a biological sensitive element (an enzyme, an antibody, DNA, cellular organelles, and cells) that is connected to an (electrochemical, optical, calorimetric, or acoustic) transformer. Using biosensors, high sensitivity, selectivity, and accuracy are reached, and they allow to carry out fast and simple measurements [1]. Biosensors are efficient and are widely used in defense and food industries, in the sphere of environment protection, but mostly in medicine [2] as a tool for diagnostics, early detection of substances causing heart diseases, for the fast detection of human papilloma virus, etc. Therefore, biosensors are intensively investigated, which stimulates the interest of scientists in them [3–5]. The family of biosensors can be divided into two groups. The first consists of a layer of receptors (an enzyme, a protein, a porphyrine, and an antigen or an antibody) sensitive to the biological material used in its construction, and the second is a layer of a conductor where a biological effect turns into a measurement (electrochemical, impedance, amperometric, optical, etc.) signal. Among a large family of biosensors, typical immunosensors are those containing a layer of sensitive and selective receptors and also an immobilized biological element, for example, an antibody, an antigen, or a hapten, i.e., immunological receptors for the molecules being investigated. In an immunsor (immunosensor), a reaction occurs based on the interaction between an antibody and an antigen or small hapten molecules. Antibodies are frequently called immunoglobulins since they are proteins connected with the immune system. Immunoglobulins are used by the immune system to identify and neutralize alien objects. They have properties of binding antigens. Antibodies and antigens can be used in the rec
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