The SIR-P Model: An Illustration of the Screening Paradox

Abstract

In previous work by this author, the screening paradox - the loss of predictive power of screening tests over time $t$ - was mathematically formalized using Bayesian theory. Where $J$ is Youden's statistic, $b$ is the specificity of the screening test and $ϕ$ is the prevalence of disease, the ratio of positive predictive values at subsequent time $k$, $ρ(ϕ_{k})$, over the original $ρ(ϕ_{0})$ at $t_0$ is given by: $ζ(ϕ_{0},k) = \frac{ρ(ϕ_{k})}{ρ(ϕ_{0})} =\frac{ϕ_k(1-b)+Jϕ_0ϕ_k}{ϕ_0(1-b)+Jϕ_0ϕ_k}$ Herein, we modify the traditional Kermack-McKendrick SIR Model to include the fluctuation of the positive predictive value $ρ(ϕ)$ (PPV) of a screening test over time as a function of the prevalence threshold $ϕ_e$. We term this modified model the SIR-P model. Where a = sensitivity, b = specificity, $S$ = number susceptible, $I$ = number infected, $R$ = number recovered/dead, $β$ = infectious rate, $γ$ = recovery rate, and $N$ is the total number in the population, the predictive value $ρ(ϕ,t)$ over time $t$ is given by: $ρ(ϕ,t) = \frac{a[\frac{βIS}{N}-γI]}{ a[\frac{βIS}{N}-γI]+(1-b)(1-[\frac{βIS}{N}-γI])}$ Otherwise stated: $ρ(ϕ,t) = \frac{a\frac{dI}{dt}}{ a\frac{dI}{dt}+(1-b)(1-\frac{dI}{dt})}$ where $\frac{dI}{dt}$ is the fluctuation of infected individuals over time $t$.