Testing is a crucial control mechanism for epidemics because it enables the health authority to detect and isolate the infected cases, thereby limiting the disease transmission to susceptible people. In this paper, an epidemic model that incorporates the testing rate as a control input is presented. The proposed model distinguishes the undetected infected from the detected infected cases, who are assumed to be isolated from the disease spreading process in the population. Two testing policies are devised: (i) best-effort strategy for testing (BEST) and (ii) constant optimal strategy for testing (COST). The BEST is a suppression policy that provides a lower bound on the testing rate to stop the growth of the epidemic. The COST is a mitigation policy that minimizes the peak of the epidemic by providing a constant, optimal allocation of tests in a certain time interval when the total stockpile of tests is limited. Both testing policies are evaluated by their impact on the number of active intensive care unit (ICU) cases and the cumulative number of deaths due to COVID-19 in France.