2021/04/15 by Balayla, Jacques
#FOS: Computer and information sciences #Methodology (stat.ME)
paper · doi:10.48550/arxiv.2104.07806
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(ϕk), over the original ρ(ϕ0) at t0 is given by: ζ(ϕ0,k) = \fracρ(ϕk)ρ(ϕ0) =(ϕk(1-b)+Jϕ0ϕk)/(ϕ0(1-b)+Jϕ0ϕk) 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 ϕe. 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) = (a[(βIS)/(N)-γI])/( a[(βIS)/(N)-γI]+(1-b)(1-[(βIS)/(N)-γI])) Otherwise stated: ρ(ϕ,t) = (a(dI)/(dt))/( a(dI)/(dt)+(1-b)(1-(dI)/(dt))) where (dI)/(dt) is the fluctuation of infected individuals over time t.