Application of differential equations with variable delay in the compartmental models of living systems

An approach to the construction of compartmental models of living systems based on linear nonautonomous differential equations with variable delay is presented. Differential equations describing the dynamics of the number of elements of the living system located in compartments are supplemented by a set of linear integral equations that reflect the dynamics of the number of elements in the process of movement between compartments. The model contains nonnegative initial data that takes into account the prehistory of the dynamics of the number of elements of the living system. The existence, uniqueness, and nonnegativity of solutions of the model on the semi-axis are established. Two-side estimates for the sum of all components of the solution are obtained. The exponential stability of the trivial solution of the system of differential equations in the absence of an external source of influx of elements of living systems is shown. A compartmental model of the dynamics of HIV-1 infection in the body of an infected person is considered. To study the model, the properties of nonsingular M-matrices are used. The conditions for exponential and asymptotic stability of the trivial solution of the model are established. The obtained relations are interpreted as the conditions for eradication of HIV-1 infection due to nonspecific factors of the human body protection.