Stability indicatrices of nonnegative matrices and some of their applications in problems of biology and epidemiology (Q2292070)

The most common structuring in problems of mathematical modeling of the dynamics of biological communities of species is structuring by age. For a linear discrete model, the dynamics of a population with an age structure is described by the Leslie model, the exponential dynamics is determined by the properties of a non-negative matrix (Leslie matrix), connecting together the state of the population at two consecutive steps in time. In this case, the presence of growth or decrease depends solely on whether the spectral radius of this matrix is more or less. To do this, it is enough to calculate the matrix stability indicatrix (an indicator of potential growth). For the Leslie matrix, this value can be constructed on the basis of a suitable representation of its characteristic polynomial, interpreted as a biological potential. It is possible to build a biological potential based on the characteristic polynomial of the original matrix for the Lefkovich and Logofet matrices. In the models defined by these matrices, the population is structured according to a certain ordered set of stages of life, passed by each individual in the direction of this order. The aim of this work was to study the possibility of constructing an indicatrix of stability for an arbitrary non-negative matrix in the context of using this indicatrix to implement the aforementioned functions of biological potential. In the case of limitations by stability problems, a large set of criteria can be used, including criteria for the non-degeneracy of \(M\)-matrices, many of which are suitable for constructing stability indicatrices within the framework indicated here. However, the positivity of major minors in degenerate cases may give a function that is not suitable for the role of an indicatrix. Therefore, the authors propose to design the simplest formulas from the matrix coefficients that are convenient for working with them in their general form. The main result is presented in the form of a theorem on the ability to construct a polynomial indicatrix for a non-negative matrix with a proof, description of the algorithm and its application. An example of the use of the stability indicatrix in the case of a matrix participating in the classical Logofet population biology model with particular cases for the Lefkovich and Leslie matrices is presented. In conclusion, a classical model of the spread of the epidemic in a distributed host population is presented. The problem of the occurrence of a pandemic and related conditions can be solved in this case by building stability indicatrices of suitable non-negative matrices.