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 $\theta$. Infected individuals go to a state "infected and tested positive" at rate $\alpha$. Both of these rates are functions of test's sensitivity and specificity. Analysis of the model permits us to derive an expression for $R_0$ and to find the conditions for reaching $R_0<1$, i.e., when the disease--free equilibrium is stable. We find that under certain conditions it is possible to get $R_0<1$, when originally, i.e., without testing, we would have $R_0>1$. We also present numerical results to cover interesting scenarios such as using different tests and to compare these results.