Fig. 3
Pathogen decay dynamics cannot be inferred from case data alone. a An outbreak of Shigella simulated with the biphasic decay disease model with parameters N= 1000, γ= 1/6, κ= 8, ρ= 0.15, V= 4E8, η= 1-(1E-5), π1= 1.1E-2, π2= 1.1E-4, ξ1= 5, ξ2= 0.2, δ1= 0.05, δ2= 0.002, and \(\mathcal {R}_{0}=\) 1.3. Biweekly case data were simulated from a binomial distribution. The monophasic decay disease model was fit to this simulated data using a binomial likelihood: parameter estimates are ακρπ= 3.4E-3 (95% CI: 2.8E-4, 4.1E-2), γ= 1.6E-1 (95% CI: 1.5E-1, 1.8E-1), τ= 6.3E-2 (95% CI: 5.2E-3, 7.6E-1), I(0)=1.0 (95% CI: 0.6,1.8). The asymptotic confidence intervals for ακρπ and τ are very wide. b Simulated pathogen decay data reveals a biphasic decay pattern and allows the estimation of the apparent fast and slow decay rates. The sum and product of these rates are represented by ξ1+δ1+ξ2+δ2= 5.24 and (ξ1+δ1)(ξ2+δ2)−δ1δ2=1.01. Using these estimates in fitting the biphasic decay disease model to the data allows us to make more accurate and precise inferences of certain parameter combinations: ακρ(ηπ1+(1−η)π2) (true: 1.09E-3, estimated: 1.18E-3 (95% CI: 1.02E-3, 1.36E-3)) and \(\mathcal {R}_{0}/N\) (true: 1.30E-3, estimated: 1.40E-3 (95% CI: 1.27E-3, 1.53E-1)). c If we can separately quantify the labile and pathogen phenotypes in the pathogen decay experiment, we can estimate ξ1=4.96, ξ2=0.23, δ1=0.06, δ2=0.0018, all close to the true values. Using the biphasic decay disease model together with these estimates, we can estimate most of the remaining model parameters: γ=0.18, κρα=0.10, η=0.72, π1=1.11E-2, π2=6.66E-4. Only η is substantially different from it’s true value. With further shedding studies, we can estimate η and then characterize the labile disease risk (\(\mathcal {R}_{0,1}\) estimated: 1.24, true: 1.30) and persistent disease risk (\(\mathcal {R}_{0,1}\) estimated: 1.5E-2, true: 3.2E-3)