The SIR-P Model: An Illustration of the Screening Paradox
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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) = ρ(ϕ_k)/ρ(ϕ_0) =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) = a[β IS/N-γ I]/ a[β IS/N-γ I]+(1-b)(1-[β IS/N-γ I]) Otherwise stated: ρ(ϕ,t) = adI/dt/ adI/dt+(1-b)(1-dI/dt) where dI/dt is the fluctuation of infected individuals over time t.
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