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

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 $\phi$ is the prevalence of disease, the ratio of positive predictive values at subsequent time $k$, $\rho(\phi_{k})$, over the original $\rho(\phi_{0})$ at $t_0$ is given by: $\zeta(\phi_{0},k) = \frac{\rho(\phi_{k})}{\rho(\phi_{0})} =\frac{\phi_k(1-b)+J\phi_0\phi_k}{\phi_0(1-b)+J\phi_0\phi_k}$ Herein, we modify the traditional Kermack-McKendrick SIR Model to include the fluctuation of the positive predictive value $\rho(\phi)$ (PPV) of a screening test over time as a function of the prevalence threshold $\phi_e$. We term this modified model the SIR-P model. Where a = sensitivity, b = specificity, $S$ = number susceptible, $I$ = number infected, $R$ = number recovered/dead, $\beta$ = infectious rate, $\gamma$ = recovery rate, and $N$ is the total number in the population, the predictive value $\rho(\phi,t)$ over time $t$ is given by: $\rho(\phi,t) = \frac{a[\frac{\beta IS}{N}-\gamma I]}{ a[\frac{\beta IS}{N}-\gamma I]+(1-b)(1-[\frac{\beta IS}{N}-\gamma I])}$ Otherwise stated: $\rho(\phi,t) = \frac{a\frac{dI}{dt}}{ a\frac{dI}{dt}+(1-b)(1-\frac{dI}{dt})}$ where $\frac{dI}{dt}$ is the fluctuation of infected individuals over time $t$.