Prior distributions, scenarios and data sources

Disease and vaccination parameters

Dynamic disease model parameters, descriptions, prior distributions, units, method used to derive the prior distribution and the type (i.e data derived, literature, assumption). All data based parameters are included. P = pulmonary TB, E = extra-pulmonary TB, v = vaccinated, i = age at vaccination, $\mathcal{U}$ = Uniform, $\mathcal{N}$ = Normal
Parameter	Description	Distribution	Units	Method	Type
$C_{\text{eff}}$	The assumed effective number of contacts per infectious TB case.	$\mathcal{N}(1, 1)$ truncated to be greater than 0.		Estimated using a dynamic model of TB transmission in England which found an effective contact rate of 1 in 1990. A conservative interval has then been applied.	Vynnycky et al.
$C^{\text{hist}}_{\text{eff}}$	The assumed historic effective number of contacts per infectious TB case.	$\mathcal{U}(C_{\text{eff}}, 20)$		Estimated using a dynamic model of TB transmission in England which found an effective contact rate of 1 in 1990 and 20 in 1901. A conservative interval has been chosen to represent the parameter uncertainty. It has been assumed that the historic contact rate is bounded below by the current contact rate.	Vynnycky et al.
$C^{\text{half-life}}_{\text{eff}}$	It is assumed that the historic effective contact rate decays from 1935 to 1980 with a half-life of C^{}_{}.	$\mathcal{N}(5, 5)$ truncated to be greater than 0.		The prior distribution is informed by historic TB notifications.	Assumption
$\beta_{\text{young-adult}}$	This parameter modifies the effective contact rate in scenarios when the transmission probability is modified for young adults (15-24).	$\mathcal{U}(0, 10)$		An uninformative prior has been used bounded above to restrict the transmission probability in young adults to be no greater than 10 times that in other age-groups.	Assumption
$\Upsilon$	The age-specific proportion of cases that have pulmonary TB	$\Upsilon_{0-14,15-59,60-89} = \mathcal{N}(0.629, 0.00101)$ , $\mathcal{N}(0.706, 0.00411)$ , $\mathcal{N}(0.750, 0.00569)$	Proportion	Estimated using the age-specific proportion of cases that had pulmonary TB in the ETS.	Derived from data
$\rho$	The age-specific proportion of pulmonary TB cases that are smear positive	$\rho_{0-14,15-59,60-89} = \mathcal{N}(0.302, 0.0189)$ , $\mathcal{N}(0.652, 0.00518)$ , $\mathcal{N}(0.536, 0.00845)$	Proportion	Estimated using the age-specific proportion of pulmonary TB cases that were smear postive in the ETS.	Derived from data
$C$	Matrix of contact rates between each age group		Non-unique yearly contacts.	For each parameter sample a contact matrix was bootstrapped from the POLYMOD survey data, standardised using the UK born population in 2005, and then averaged to provided a symmetric contact matrix.	Mossong et al.
$\iota(t)$	The age-specific number of non-UK born pulmonary TB cases in England each year		Cases	The number of pulmonary non-UK born cases for each year were extracted from the ETS and grouped by age.	Derived from data
$\iota_{\text{scale}}$	Scaling parameter for the importation of non-UK born cases between 1960 and 2000.	$\mathcal{N}(0, 200)$		This largely uninformative prior range was chosen so that non-UK born scaling can vary between approximately linear to approximately constant.	Assumption
$M$	The proportion of mixing between the UK born and non-UK born population.	$\mathcal{N}(1, 1)$ truncated to be greater than 0	Proportion	Any degree of mixing is allowed as there is little data on which to base this estimate. Mixing greater than 1 is allowed as this is used to represent non-UK born cases being in some way more infectious than non-UK born cases.	Assumption
$M_{\text{young-adult}}$	This parameter modifies the non-UK born mixing rate in scenarios when $M$ is modified for young adults (15-24).	$\mathcal{U}(0, 10)$		An uninformative prior has been used bounded above to restrict non-UK born mixing in young adults to be no greater than 10 times that in other age-groups.	Assumption
$\chi$	Age-specific protection from infection with TB due to BCG vaccination	$\chi^v_{i} = \mathcal{N}(0.185$ , $0.0536)$ , where $i$ is the age group vaccinated.	Proportion	A meta-analysis of the protection from infection due to BCG vaccinatiion in children. It has been assumed that there is no reduction in protection in UK born adults.	Roy et al.
$\epsilon_H$	The age-specific rate of transition to active disease during high risk latent period.	$\epsilon_H^{0-4,5-14,15-89} = \mathcal{N}(0.00695, 0.00130)$ , $\mathcal{N}(0.0028, 0.000561)$ , $\mathcal{N}(0.000335, 0.0000893)$	$days^{-1}$	From fitting a similar model to contact data in Australia, and Holland. Distribution derived by the assumption of a normal distribution based on 95% credible intervals.	Ragonnet et al.
$\kappa$	The reciprocal of the age-specific average high risk latent period.	$\kappa^{0-4,5-14,15-89} = \mathcal{N}(0.0133, 0.00242)$ , $\mathcal{N}(0.0120, 0.00207)$ , $\mathcal{N}(0.00725, 0.00191)$	$days^{-1}$	From fitting a similar model to contact data in Australia, and Holland. Distribution derived by the assumption of a normal distribution based on 95% credible intervals.	Ragonnet et al.
$\epsilon_L$	The reciprocal of the age-specific average low risk latent period.	$\epsilon_L^{0-4,5-14,15-89} = \mathcal{N}(8.00e^{-6}, 4.08e^{-6})$ , $\mathcal{N}(9.84e^{-6}, 4.67e^{-6})$ , $\mathcal{N}(5.95e^{-6}, 2.07e^{-6})$	$days^{-1}$	From fitting a similar model to contact data in Australia, and Holland. Distribution derived by the assumption of a normal distribution based on 95% credible intervals.	Ragonnet et al.
$\epsilon^{\text{older-adult}}_L$	This parameter modifies the activation rate of low risk latent cases who are 70+ and reduces the activaton rate for other adults so that the mean activation rate is $\epsilon_L^{15-89}$	$\mathcal{N}(2, 0.5)$ truncated to be at least 1.		Evidence suggests that activation risk increases when individuals enter old age. A largely uninformative prior has been used centred around an increase in risk of double the average rate.	Horsburgh, Jr. et al.
$\alpha_i^T$	The BCG vaccine effectiveness at preventing the development of active TB disease in a TB free population	$\alpha^T_{i,i+5,i+10,i+15,i+20,i+25} = 1 - e^{\alpha^{\text{ln}(T)}_{i,i+5,i+10,i+15,i+20,i+25}}$ , where $\alpha^{\text{ln}(T)}_{i,i+5,i+10,i+15,i+20,i+25} = \mathcal{N}(-1.86, 0.22)$ , $\mathcal{N}(-1.19, 0.24)$ , $\mathcal{N}(-0.84, 0.22)$ , $\mathcal{N}(-0.84, 0.2)$ , $\mathcal{N}(-0.28, 0.19)$ , $\mathcal{N}(-0.23, 0.29))$ and $i$ is the age group vaccinated	Proportion	Poisson regression used to calculate Risk Ratios from literature values. A distribution is then found using the log normal approximation. Effectiveness estimates are caculated using 1 minus the exponentiated log normal distribution.	Hart et al. and Mangtani et al.
$\delta$	Reduction in susceptibilty to infection for low risk latent cases.	$\mathcal{N}(0.78$ , $0.0408)$	Proportion	A review of prospective cohort studies of persons exposed to individuals with infectious tuberculosis that was published prior to the widespread treatment of latent tuberculosis.	Andrews et al.
$\nu^{P, E}$	The reciprocal of the average infectious period	$\nu^P_{(0-14,15-89)} = \mathcal{N}(0.181, 0.310)^{-1}$ , $\mathcal{N}(0.328, 0.447)^{-1}$ , $\nu^E_{(0-14, 15-89)} = \mathcal{N}(0.306, 0.602 )^{-1}$ , $\mathcal{N}(0.480, 0.866)^{-1}$	$years^{-1}$	Estimated based on the time from initial symptoms to starting treatment	Derived from data
$\phi$	The reciprocal of the time to succesful treatment completion	$\phi_{0-14,15-69,70-89} = \mathcal{N}(0.606,0.237)^{-1}$ , $\mathcal{N}(0.645, 0.290)^{-1}$ , $\mathcal{N}(0.616, 0.265)^{-1}$ and truncated to be greater than 4 months	$years^{-1}$	Estimated based on the time from starting treatment to treatment completion.	Derived from data
$\mu$	Rate of age-specific pulmonary/extra-pulmonary TB TB mortality	$\mu_{0-14,15-59,60-89} = \mathcal{N}(0.0039, 0.018)$ , $\mathcal{N}(0.0226, 0.00787)$ , $\mathcal{N}(0.117, 0.0165)$ truncated to be greater than 0.	$years^{-1}$	Estimated based on outcomes at 12 months where cause of death was known, including all-cause deaths in the denominator.	Derived from data
$\zeta$	Rate of loss to follow up	$\zeta_{0-14,15-59,60-89} = \mathcal{N}(0.00976, 0.0179)$ , $\mathcal{N}(0.0304, 0.00764)$ , $\mathcal{N}(0.00614, 0.0159)$ , truncated to be greater than 0.	$years^{-1}$	Estimated based on outcomes at 12 months for TB cases	Derived from data

Demographic model parameters

Demographic model parameters, descriptions, prior distributions, units, method used to derive the prior distribution and the type (i.e data derived, literature, assumption). $\mathcal{U}$ = Uniform and i = age at vaccination.
Parameter	Description	Distribution	Units	Method	Type
$\omega(t)$	Time varying births			The dataset contains the estimated number of births from 1929-2015 in England. From 2016 onwards the numbers of births are projections as published by ONS.	Derived from data
$\gamma$	BCG vaccination coverage	$\gamma_{i} = \mathcal{N}(0.75, 0.05)$ Where $i$ is the age group vaccinated.	Proportion	England has a robust national health service and an established system for providing BCG vaccination.	Assumption
$\theta$	Rate of ageing		$years^{-1}$	Defined as the reciprocal of the width of the modelled age groups.	Model defined
$\mu^{\text{all-cause}}(t)$	Time varying all-cause age-specific mortality rate		$years^{-1}$	Age specific mortality averaged across age group from 1981-2015. From 2016 onwards, and prior to 1981, mortality rates are modelled using a exponential model fit to data from 1981 until 2015.	Derived from data

Data sources

Sources used to parameterise the disease and demographic models. Parameters that use the source are given, as well as the study type, setting, year/years studied and a description of the study/data source.
Parameters	Study Type	Setting	Year	Description	Source
$\iota(t)$ , $\mu$ , $\nu^{P, E}$ , $\phi$ , $\rho$ , $\Upsilon$ , $\zeta$	-	England	2000-2015	The Enhanced Tuberculosis Surveillance System (ETS) is a robust national data collection system that collects demographic and microbiological data on all notified cases in England.	ETS
$\mu^{\text{all-cause}}(t)$ , $\omega(t)$	-	England	-	The Office for National Statistics (ONS) compiles demographic, health, enconomic, and social data for the United Kingdom	ONS
$C_{\text{eff}}$ , $C^{\text{hist}}_{\text{eff}}$	Dynamic modelling study	England	Up to 1990	Used a dynamic model of tuberculosis, robustly parameterised to the available evidence and including realistic population demographis to estimate the effective contact rate of TB over time until the 1990’s in the UK born white male population.	[@Vynnycky1999]
$C$	Contact survey	Europe - including the United Kingdom	2005	Conducted contact surveys, based on a contact diary, in multiple European countries. Contacts were stratified by age and type of contact. In the United Kingdom over a thousand people were surveyed.	[@Mossong2008]
$\chi$	Systematic review and meta-analysis	Global	Up to 2014	A meta-analysis; conducted with the aim of determining whether BCG vaccination protects against Mycobacterium tuberculosis infection as assessed by interferon $\gamma$ release assays (IGRA) in children. Estimated both protection from initial latent infection and active TB disease.	[@Roy2014]
$\epsilon_H$ , $\epsilon_L$ , $\kappa$	Systematic review	Global	Up to 2017	Aimed to determine which dynamic TB model structure best captured the observed activation dynamics of TB. Identified 6 different commonly used model structures and compared them by fitting to activation data from the Netherlands and Australia.	[@Ragonnet2017]
$\alpha_i^T$	Clinical trial	England	1950-1965	Investigated the effectiveness of the BCG vaccine at preventing TB disease when given at what as then school-leaving age. Followed the cohort over 15 years and estimated the effectiveness of the BCG vaccien in 2.5 year intervals from vaccination.	[@Hart1972]
$\alpha_i^T$	Population based case-control study	England	2002-2014	Recruited UK-born White subjects with TB and randomly smapled White community controls. Cox regression was used to adjusted for known confounders and the effectiveness of the BCG vaccine was estimated from 10 years after vaccination until 30 years after vaccinaton.	[@Mangtani2017]
$\delta$	Systematic review and meta-analysis	Global	Up to 2012	Reviewed prospective cohort studies of persons exposed to individuals with infectious TB. Only included studies that were published before the widespread treatment of latent TB. Aimed to estimate the reduction in re-infection for latent TB cases.	[@Andrews2012]
$\beta_{\text{young-adult}}$ , $\gamma$ , $M_{\text{young-adult}}$ , $M$	-	England	-	Where data, or literature, sources were not available assumed values were used based on expert opinion	Assumption

Scenarios

Summary of planned scenario analyses to be carried out in the next chapter as part of model fitting by comparision of the goodness of fit to the data.
Parameter	Scenario
$\beta_a$ - transmission probability	Constant across all age groups
	Variable in young adults (15-29)
$M$ - Non-UK born mixing	Constant across all age groups
	Variable in young adults (15-29)

Sam Abbott

2019-07-31

Disease and vaccination parameters

Demographic model parameters

Data sources

Scenarios

Contents