PROBLEM OF OPTIMAL CONTROL OF EPIDEMIC IN VIEW OF LATENT PERIOD

Abstract

The problem of optimal control of epidemic through vaccination and isolation, taking into account latent period is considered. The target function is minimized-functionality summarizing costs on epidemic prevention and treatment and also considering expenses on infected people left at the end of control T who may be a new source of epidemic. On the left endpoint of the integration segment initial data is given-quantity of infected and confirmed people at the moment t, the right endpoint is free. The dynamic constraints are written by way of a system of simple differential equations describing the speed of changes of number of subjected to infection and number of already infected. Besides the inhomogeneous community is considered, consisting of four age groups (babies, preschool children, school children and adults). The speed of vaccination (number of vaccinated per a time unit) and isolation speed are used as the control functions. There are some restrictions on control above and below. The latent period is described by the constant h and is part of the equation describing the contamination speed of people as a retarding in argument t, i.e. a person being in a latent period infects others not being aware of his disease. For problem solving Pontryagin maximum principle is used where it can be seen that the control is piecewise constant. The result of numerical implementation of discrete problem of optimal control is given. The conclusions are made that the latent period significantly influence the incidence rate and as consequence the costs on epidemic suppression. The programme based on the programming language Delphi gives an opportunity to estimate the scale of epidemic at different initial data and restrictions on control as well as to find an optimal control minimizing costs on epimedic suppression.