  Motivated with various responses of world governments to COVID-19, here we
develop a toy model of the dependence epidemics spreading on the availability
of tests for disease. Our model, that we call SUDR+K, is based on usual SIR
model, but it splits the total fraction of infected individuals into two
components: those that are undetected and those that are detected through
tests. Moreover, we assume that available tests increase at a constant rate
from the beginning of epidemics but are consumed to detect infected
individuals. Strikingly we find a bi-stable behavior between a phase with a
giant fraction of infected and a phase with a very small fraction. We show that
the separation between these two regimes is governed by a match between the
rate of testing and a rate of infection spread at given time. We also show that
the existence of two phases does not depend on the mathematical choice of the
form of the term describing the rate at which undetected individuals are tested
and detected. Presented research implies that a vigorous early testing
activity, before the epidemics enters into its giant phase, can potentially
keep epidemics under control, and that even a very small change in rate of
testing can increase or decrease the size of the whole epidemics of various
orders of magnitude. For the real application of realistic model to ongoing
epidemics, we would gladly collaborate with field epidemiologists in order to
develop quantitative models of testing process.
