Appendix 1—table 1. Model parameters.

Parameter	Description
${\displaystyle p_T}$	Standard SICCT Sensitivity. Probability of positive tuberculin test for ${\displaystyle R,I}$ individuals at standard definition.
${\displaystyle 1-p_{FP}}$	Standard SICCT Specificity. Probability of negative tuberculin test for ${\displaystyle S,O,R,I}$ individuals at standard definition (1 - probability of a false positive ${\displaystyle p_{FP}}$).
${\displaystyle p_T^{\prime }}$	Severe SICCT Sensitivity. Probability of positive tuberculin test for ${\displaystyle R,I}$ individuals at severe definition.
${\displaystyle 1-p_{FP}^{\prime }}$	Severe SICCT Specificity. Probability of negative tuberculin test for ${\displaystyle S,O,R,I}$ individuals at severe definition. (${\displaystyle 1-}$ probability of false positive ${\displaystyle p_{FP}^{\prime }}$).
${\displaystyle p_{RI}}$	Slaughterhouse detection. Relative sensitivity of finding lesioned or culture positive animals (${\displaystyle O,O_{V1},O_{V2},R,R_V,I,I_V}$ status) under routine inspection compared to reactor inspection.
${\displaystyle T_O}$	Occult Period. Mean length of time that animals are undetectable (occult) to SICCT test.
${\displaystyle T_R}$	Reactive Period. Mean length of time between infection and animals becoming infectious.
${\displaystyle \beta }$	Transmission parameter associated with density dependence (rate per day, dimensions change with $q$).
${\displaystyle q}$	Transmission parameter measuring the strength of density dependence (range 0–1)
${\displaystyle {\chi }_1}$	Transmission parameter measuring infectious pressure per susceptible per year in PTI 1
${\displaystyle {\chi }_2}$	Transmission parameter measuring infectious pressure per susceptible per year in PTI 2
${\displaystyle {\chi }_4}$	Transmission parameter measuring infectious pressure per susceptible per year in PTI 4
${\displaystyle H}$	Herd size. Sampled from empirical distribution and maintained constant through individual simulations
${\displaystyle H_M=165}$	Constant equal to mid-point of range of herd sizes within study population. Used to transform density dependence of force of infection.
${\displaystyle p_D}$	DIVA sensitivity Probability of positive DIVA test in infected vaccinates ${\displaystyle (O_{V1},O_{V2},R_V,I_V)}$.
${\displaystyle 1-p_{DFP}}$	DIVA specificity Probability of negative DIVA test for uninfected vaccinates ${\displaystyle V_1,V_2}$ ($V_1,V_2$ probability of a false positive ${\displaystyle p_{DFP}}$)
${\displaystyle {\epsilon }_S}$	Vaccine Efficacy Reduction in risk of infection for susceptible vaccinates (status ${\displaystyle V_1}$)
${\displaystyle {\epsilon }_I}$	Vaccine Efficacy Reduction in risk of infectiousness for infected vaccinates ${\displaystyle (R_V,I_V)}$ for SORI model, and in addition ${\displaystyle \left(O_{V1},O_{V2}\right)}$ for SOR model.
${\displaystyle T_V}$	Protective Period. Mean length of time that animals are protected by BCG vaccination (status ${\displaystyle V_1}$)