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\(C_{\text{eff}}\)
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The assumed effective number of contacts per infectious TB case.
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\(\mathcal{N}(1, 1)\) truncated to be greater than 0.
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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.
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Vynnycky et al.
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\(C^{\text{hist}}_{\text{eff}}\)
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The assumed historic effective number of contacts per infectious TB case.
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\(\mathcal{U}(C_{\text{eff}}, 20)\)
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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.
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Vynnycky et al.
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\(C^{\text{half-life}}_{\text{eff}}\)
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It is assumed that the historic effective contact rate decays from 1935 to 1980 with a half-life of C^{}_{}.
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\(\mathcal{N}(5, 5)\) truncated to be greater than 0.
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The prior distribution is informed by historic TB notifications.
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Assumption
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\(\beta_{\text{young-adult}}\)
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This parameter modifies the effective contact rate in scenarios when the transmission probability is modified for young adults (15-24).
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\(\mathcal{U}(0, 10)\)
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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.
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Assumption
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\(\Upsilon\)
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The age-specific proportion of cases that have pulmonary TB
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\(\Upsilon_{0-14,15-59,60-89} = \mathcal{N}(0.629, 0.00101)\), \(\mathcal{N}(0.706, 0.00411)\), \(\mathcal{N}(0.750, 0.00569)\)
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Proportion
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Estimated using the age-specific proportion of cases that had pulmonary TB in the ETS.
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Derived from data
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\(\rho\)
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The age-specific proportion of pulmonary TB cases that are smear positive
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\(\rho_{0-14,15-59,60-89} = \mathcal{N}(0.302, 0.0189)\), \(\mathcal{N}(0.652, 0.00518)\), \(\mathcal{N}(0.536, 0.00845)\)
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Proportion
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Estimated using the age-specific proportion of pulmonary TB cases that were smear postive in the ETS.
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Derived from data
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\(C\)
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Matrix of contact rates between each age group
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Non-unique yearly contacts.
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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.
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Mossong et al.
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\(\iota(t)\)
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The age-specific number of non-UK born pulmonary TB cases in England each year
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Cases
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The number of pulmonary non-UK born cases for each year were extracted from the ETS and grouped by age.
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Derived from data
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\(\iota_{\text{scale}}\)
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Scaling parameter for the importation of non-UK born cases between 1960 and 2000.
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\(\mathcal{N}(0, 200)\)
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This largely uninformative prior range was chosen so that non-UK born scaling can vary between approximately linear to approximately constant.
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Assumption
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\(M\)
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The proportion of mixing between the UK born and non-UK born population.
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\(\mathcal{N}(1, 1)\) truncated to be greater than 0
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Proportion
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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.
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Assumption
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\(M_{\text{young-adult}}\)
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This parameter modifies the non-UK born mixing rate in scenarios when \(M\) is modified for young adults (15-24).
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\(\mathcal{U}(0, 10)\)
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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.
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Assumption
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\(\chi\)
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Age-specific protection from infection with TB due to BCG vaccination
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\(\chi^v_{i} = \mathcal{N}(0.185\), \(0.0536)\), where \(i\) is the age group vaccinated.
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Proportion
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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.
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Roy et al.
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\(\epsilon_H\)
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The age-specific rate of transition to active disease during high risk latent period.
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\(\epsilon_H^{0-4,5-14,15-89} = \mathcal{N}(0.00695, 0.00130)\), \(\mathcal{N}(0.0028, 0.000561)\), \(\mathcal{N}(0.000335, 0.0000893)\)
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\(days^{-1}\)
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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.
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Ragonnet et al.
|
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\(\kappa\)
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The reciprocal of the age-specific average high risk latent period.
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\(\kappa^{0-4,5-14,15-89} = \mathcal{N}(0.0133, 0.00242)\), \(\mathcal{N}(0.0120, 0.00207)\), \(\mathcal{N}(0.00725, 0.00191)\)
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\(days^{-1}\)
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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.
|
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\(\epsilon_L\)
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The reciprocal of the age-specific average low risk latent period.
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\(\epsilon_L^{0-4,5-14,15-89} = \mathcal{N}(8.00e^{-6}, 4.08e^{-6})\), \(\mathcal{N}(9.84e^{-6}, 4.67e^{-6})\), \(\mathcal{N}(5.95e^{-6}, 2.07e^{-6})\)
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\(days^{-1}\)
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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.
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Ragonnet et al.
|
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\(\epsilon^{\text{older-adult}}_L\)
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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 \(\epsilon_L^{15-89}\)
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\(\mathcal{N}(2, 0.5)\) 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.
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Horsburgh, Jr. et al.
|
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\(\alpha_i^T\)
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The BCG vaccine effectiveness at preventing the development of active TB disease in a TB free population
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\(\alpha^T_{i,i+5,i+10,i+15,i+20,i+25} = 1 - e^{\alpha^{\text{ln}(T)}_{i,i+5,i+10,i+15,i+20,i+25}}\), 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
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Proportion
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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.
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Hart et al. and Mangtani et al.
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\(\delta\)
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Reduction in susceptibilty to infection for low risk latent cases.
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\(\mathcal{N}(0.78\), \(0.0408)\)
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Proportion
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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.
|
|
\(\nu^{P, E}\)
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The reciprocal of the average infectious period
|
\(\nu^P_{(0-14,15-89)} = \mathcal{N}(0.181, 0.310)^{-1}\), \(\mathcal{N}(0.328, 0.447)^{-1}\), \(\nu^E_{(0-14, 15-89)} = \mathcal{N}(0.306, 0.602 )^{-1}\), \(\mathcal{N}(0.480, 0.866)^{-1}\)
|
\(years^{-1}\)
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Estimated based on the time from initial symptoms to starting treatment
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Derived from data
|
|
\(\phi\)
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The reciprocal of the time to succesful treatment completion
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\(\phi_{0-14,15-69,70-89} = \mathcal{N}(0.606,0.237)^{-1}\), \(\mathcal{N}(0.645, 0.290)^{-1}\), \(\mathcal{N}(0.616, 0.265)^{-1}\) 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
|