This is a theoretical study of the SIR model — a popular mathematical model of the propagation of infectious diseases. We construct a solution of the Cauchy problem for a system of two ordinary differential equations describing in integral form the concentration dynamics of infected and recovered individuals in an immune population. A qualitative analysis is carried out of the stationary system states using the Lyapunov function. An expression is obtained for the coordinates of the equilibrium points in terms of the Lambert W-function for arbitrary initial values. The application of the SIR model for the description of COVID-19 propagation dynamic is demonstrated.

An $$n$$-dimensional economic model is considered that has a Cobb–Douglas production function on the infinite planning horizon such that the utility function is an integral-type functional with a discount and a logarithm-type integrant. It is assumed that all of the model’s amortization factors are equal to one another. The constructed optimum control contains $$n-1$$ special segments that are described analytically. A special sweep procedure for consecutively solving two Cauchy problems on each segment is developed to find moments of switching between segments and the shape of the optimum trajectory. On the last segment, the optimum trajectory lies along a special ray; from the viewpoint of economy, this ray can be interpreted as the mode of equilibrium growth. A Pontrjagin maximum-principle problem with a special transversality condition is used to construct the optimum solution. Optimality is confirmed using the Kiselev theorem on sufficient conditions. Moving to problems of large dimensions considerably increases the number of technical difficulties. The description of the optimality verification procedure is therefore presented in detail for methodological reasons.

We investigate the allocation of resources in a two-sector economic model with a Cobb–Douglas production function for different depreciation rates with an integral functional on a finite time horizon. The problem is reduced to some canonical form by the scaling of phase variables and time. The extremal solution constructed by Pontryagin’s maximum principle is shown to be optimal. For a sufficiently large planning horizon, the optimal control has two or three switch points, contains one singular section, and vanishes on the final section. A transition “calibration” regime is observed between the singular section, where the system moves along a singular ray, and the final section. The maximum-principle boundary-value problem is solved in explicit form and the solution is illustrated with graphs based on numerical calculations.

ТРУДЫ ИНСТИТУТА МАТЕМАТИКИ И МЕХАНИКИ УрО РАН

The classical two-dimensional Fuller problem is considered. The boundary value problem of Pontryagin’s maximum principle is considered. Based on the central symmetry of solutions to the boundary value problem, the Pontryagin maximum principle as a necessary condition of optimality, and the hypothesis of the form of the switching line, a solution to the boundary value problem is constructed and its optimality is substantiated. Invariant group analysis is in this case not used. The results are of considerable methodological interest.

We consider the resource allocation problem in a two-sector economy with a Constant Elasticity of Substitution (CES) production function on a given sufficiently long finite planning horizon. The performance criterion being optimized is one of the phase coordinates at the terminal time. The scalar control u satisfies the geometrical constraint u ∈ [0, 1]. The optimal control contains a singular section. Analyzing the Pontryagin maximum principle boundary-value problem we find the extremal triple and prove its optimality using a special representation of the functional increment (the theorem of sufficient conditions of optimality in terms of maximum-principle constructs). The solution is constructed with the aid of a Lambert special function y = W(x), that solves the equation ye y = x. The optimal control consists of three sections: the initial section that involves motion toward the singular ray L sng = {x 1 = x 2 > 0} , the singular section that involves motion along the singular ray, and terminal section that involves motion under zero control. When the initial state is on the singular ray, the optimal control consists of two sections — singular and terminal. The article is related to a number of the authors’ publications using other production function.

An infinite-horizon two-sector economy model with a Cobb–Douglas production function is studied for different depreciation rates, the utility function being an integral functional with discounting and a logarithmic integrand. The application of the Pontryagin maximum principle leads to a boundary value problem with special conditions at infinity. The presence of singular modes in the optimal solution complicates the search for a solution to the boundary value problem of the maximum principle. To construct the solution to the boundary value problem, the singular modes are written in an analytical form; in addition, a special version of the sweep algorithm in continuous form is proposed. The optimality of the extremal solution is proved.

An n-dimensional problem of optimal economic growth in a multifactor model with the Cobb–Douglas production function and an integral-type functional with discounting is investigated. The model is studied by assuming that all amortization coefficients are equal. A constructive description of an optimal solution for a sufficiently large planning horizon and a sufficiently small discount coefficient is obtained. The extremal solution is described in analytical form. The studied problem with other production functions has a biological interpretation in an optimal growth model of agricultural plants with n vegetative organs during a specific finite time interval.

In this work, we study a two-sector economic model with the Cobb–Douglas production function on an infinite planning horizon where the utility function is a functional of an integral form and a Lagrangian of a logarithmic type. A one-dimensional equation is obtained that depends only on the coefficients of elasticity and amortization, and determines the possible special modes. The special modes are described in analytical form.

We investigate a one-dimensional nonlinear optimal control problem and describe its optimal trajectory and optimal control for three distinct cases.

We study the resource allocation problem in a two-sector economic model with a two-factor Cobb–Douglas production function for various amortization factors on a finite time horizon with a functional of the integral type. The problem is reduced to a canonical form by scaling the state variables and time. We show that the extremal solution constructed with the use of the maximum principle is optimal. For a sufficiently large planning horizon, the optimal control has two or three switching points, contains one singular segment, and is zero on the terminal part. The considered problem with different production functions admits a biological interpretation in a model of balanced growth of plants on a given finite time interval.