Imperfect Testing of Individuals for Infectious Diseases: Mathematical Model and Analysis

Testing symptomatic individuals for a disease can deliver treatment resources, if tests’ results turn positive, which speeds up their treatment and might also decrease individuals’ contacts to other ones. An imperfect test, however, might incorrectly consider susceptible individuals to be infected (false positives). In this case, testing reduces the epidemic in the expense of potentially misclassifying individuals. We present a mathematical model that describes the dynamics of an infectious disease and its testing. Susceptible individuals turn to “susceptible but deemed infected” at rate θ. Infected individuals go to a state “infected and tested positive” at rate α. Both rates are functions of test’s sensitivity and specificity. Analysis of the model permits us to derive an expression for R0 and to find the conditions for reaching R0 < 1, i.e., when the disease–free equilibrium is stable. We also present numerical results to cover interesting scenarios such as using different tests and to compare these results. We find for different sensitivity and specificity values the conditions permitting to get the basic reproduction number R0 < 1, when originally, i.e., without testing, we would have R0 > 1. We also find for a given sensitivity and specificity, the critical testing rate for reaching R0 < 1.