Conceptual model for relationship between test positivity, prevalence of infection, and testing rate.

(A) Compartmental representation of how the relationships between new infections, undiagnosed and diagnosed prevalence (I_U and I_D) and seroprevalence (SP_U and SP_D) are modeled for each state, given a bias with power n. All observational inputs are the past τ-day averages of number of positive tests N_+,τ(t) and number of tests performed N_test,τ(t), the corresponding test positivity rate P_+,τ(t) and reported case rate C_+,τ(t), and the state population size N. For diagnosed prevalence and seroprevalence, the observational input is the daily reported cases N_+,τ, and the model parameters are the recovery time after diagnosis T_rec and the time from infection to seropositivity T_inf. For undiagnosed prevalence and seroprevalence, our model assumes the test positivity rate is correlated to delayed undiagnosed disease prevalence with a bias parameter b(t) modeled as a negative power function of the testing rate b(t) = [N_test,τ(t)/N]^–n (Eq 2). The additional parameters consist of the power parameter n and the initial (missed) seroprevalence SP_o. The effective rate parameter 1/T_eff is time-dependent, and accounts for both T_inf and ongoing diagnoses so as to not “double count.” Prevalence and seroprevalence are evaluated with a lag time t_lag, assumed equal to half the averaging time τ/2. In (B), the diagonal lines represent different values of the bias parameter. In (C), the relationship between testing rate and bias parameter represented by Eq (4) is illustrated. Here the shaded region represents different powers n ranging from 0.1 (lower bound bias) to 0.9 (upper bound bias), the solid line represents n = ½.