Table 3.

Estimates of the reproductive number from mathematical models with empirical data

Name of first author [publication]	Location, time of data	Type of data	Objective	Methods	Assumptions	Reproductive number type	Estimated reproductive number
Zhao [34]	China, 2005–2016	CDC data	To investigate the impact of age on TB transmission	SEIR model with age structure; use least squares to get parameters that align with TB data in China; use Latin hypercube sampling to get CI	Although the susceptible compartment was stratified by age, the other compartments were not age-stratified thus assuming no difference in age for those compartments	Basic	1.786 (95% CI 1.775–1.796)
Liu [35]	China, 2004–2014	Annual TB case data	To use modelling to investigate the impact of different vaccination strategies (constant or pulse BCG) on TB transmission	Compartmental models with vaccination compartments	Assumptions made for all parameter values	Basic	1.19
Yang [36]	Shaanxi, China, 2004–2012	Notifiable active TB cases by month	Study the seasonality impact on TB transmission dynamics	A seasonality TB compartmental model: subjects either entered latent or diseased compartment; contact rate, reactivation rate and disease-induced death rate are periodic continuous functions	Parameter values for recruitment rate, natural death rate, recovery rate	Basic	Dependent on parameter values
Nebenzahl-Guimaraes [28]	The Netherlands, 1993–2011	Surveillance and RFLP data	Determine if mycobacterial lineages affect infection risk, clustering and disease progression among Mycobacterium tuberculosis cases	Descriptive and regression approach; DNA fingerprinting to link cases	All secondary cases captured in surveillance data; genetic matching accurately reflects transmission patterns	Effective	Range: 0.17–1.04
Narula [37]	India, 2006–2011	Quarterly reported data from Central TB Division	Estimate basic R0 for TB	Compartmental model with Bayesian melding technique to estimate parameters; Susceptible, latent, infected compartments instead of SIR	Some parameter values assumed with reference in the differential equations	Basic	0.92, averaged for India overall with range 0.72–0.98
Zhang [38]	China, 2005–2012	Monthly case reporting data from CDC	Estimate effective R0 of TB by year	Compartmental model adding hospitalised compartment; Chi-square test for optimal parameters	An upper bound for number of initially susceptible people, natural death rate, initial number of latent individuals	Effective	Range from 3.318 to 4.302 from year 2005 to 2012
Ypma [39]	The Netherlands, 1993–2007	RFLP data	Explore the high heterogeneity in the number of secondary cases caused per infectious individual for TB	Model ‘superspreading’ parameter as a negative binomial distribution	Immigrants who have been in the country for less than 6 months at diagnosis are index cases themselves	Fingerprint reproductive number as a function of the effective reproductive number and the probability that the fingerprint of the infected person is different than its infector	0.48 (95% CI 0.44–0.59)
Andrews [40]	Cape Town, South Africa, 2011	Carbon dioxide data, public transit usage data from national survey	Estimate risk of TB transmission on 3 modes of public transit	Modified Wells-Riley model for airborne disease transmission	Duration of infectiousness of 1 year; used TB and HIV parameters from studies in the same area; natural history parameters from the literature	Basic	Dependent on duration of infectiousness and frequency of transit usage
Okuonghae [41]	Benin city, Nigeria, 2008	Survey data	Assess how control strategies on addressing TB transmission parameters can minimise incidence	Compartmental model adding compartments of disease awareness level, identified infectiousness	Model parameter values such as recruitment rate, recovery rate from the literature	Basic, under treatment	Dependent on parameter values
Liao [42]	Taiwan, 2005–2010	Monthly data from CDC	Estimate MDR-TB infection risk	Mathematical probabilistic two-strain model with compartments for drug-sensitive and drug-resistant subjects; dose–response model for relationship between R0 and total proportion of infected population	Some model parameter values from data, some from the literature; assumed 0.99 of people latently infected were drug sensitive and 0.01 were drug resistant	Basic	Hwalien County: 0.89 (95% CI 0.23–2.17) for drug sensitive; 0.38 (95% CI 0.05–1.30) for multi-drug resistant; Taitung County: 0.94 (95% CI 0.24–2.28) for drug sensitive; 0.38 (95% CI 0.05–1.33) for multi-drug resistant; Pingtung County: 0.85 (95% CI 0.21–2.08) for drug sensitive; 0.34 (95% CI 0.04–1.13) for multi-drug resistant; Taipei City: 0.84 (95% CI 0.21–2.00) for drug sensitive; 0.30 (95% CI 0.04–0.97) for multi-drug resistant;
Liao [43]	Taiwan, 2004–2008; selected three areas with the highest incidence, one with the lowest incidence	Monthly disease burden TB data from Taiwan CDC	Examine TB population dynamics and assess potential infection risk	Compartmental model with susceptible, latently infected, infectious, non-infectious and recovered compartments; incorporated reactivation, relapse and reinfection	Some parameter values taken from the literature, some estimated from data	Basic, estimated as sum of fast, slow and relapse	Highest R0 total in Hwalien: 1.65 with 95th percentile range 0.45–6.45; Taipei lowest at 1.5 (0.45–4.98); Taitung: 1.72; Pingtung: 1.65
Liu [44]	China, 2000–2008	Data from the National Bureau of Statistics	Incorporate migration to study TB transmission	SEIR compartments for rural residents, migrant workers and urban population	Model parameters calculated from website data; migration rates	Basic	No explicit estimate
Borgdorff [29]	The Netherlands, 1993–2007	RFLP data	Determine to what extent tuberculosis trends in the Netherlands depend on secular trend, immigration and recent transmission	DNA fingerprinting to link cases	All secondary cases captured in surveillance data; genetic matching accurately reflects transmission patterns	Basic	0.24 (95% CI 0.21–0.26)
Liu [45]	China, Jan, 2005–Dec, 2008	Monthly notification data from Ministry of Health	Develop a model incorporating seasonality and define basic reproduction ratio	Used periodic infection rate and reactivation rate to incorporate seasonality in the compartmental model; considered fast and slow progression	Parameters such as recruitment rate, natural death rate were assumed to be constants; some parameter values assumed and some taken from the literature	Basic	Dependent on parameter values with range 0.4–2.6
Brooks-Pollock [46]	Ukraine, 1959 and 2006	Mortality data	Explore the effect of age structure on TB infection and disease prevalence, basic reproductive number and impact of intervention	Basic SEIR mathematical model with assumptions about survivorship	A survivorship function which could be described in terms of age and life expectancy	Basic	Dependent on progression rate with range 0–0.85
Basu [47]	KwaZulu-Natal, South Africa	Extensively drug-resistant TB data (XDR-TB)	Model XDR-TB transmission dynamics	Model XDR-TB incorporating the existing XDR detection rate and treatment system	Even mixing of air; range of key parameters in the model	Effective	1.97, range 0.7–4.6; 1.23, range 0.4–3.1 when combining screening and therapy; 1.38, range 0.6–3.3 with South African strategic plan alone.
Furuya [7]	Japan, 2000–2005	Exposure data	Quantify the risk of TB infection in an internet café where people without homes stayed overnight	Wells-Riley model to estimate the reproductive number	Patients stayed in a confined space for 150 days; some values in the Wells-Riley equation assumed, others from the literature	Estimated as a function of exposure period	Dependent on exposure period
Long [9]	Southern India, 2004–2006	HIV-TB co-epidemics data	Model HIV-TB co-epidemics and explore hypothetical treatment effect	First model: susceptibility to either or both diseases compartments; second model: SII*SEI	A linear relationship between treatment levels and the associated parameters; model parameters from the literature	Basic	R = 3.55 when no active treatment for TB
Borgdorff [48]	The Netherlands, 1995–2002	RFLP data	Assess progress towards TB elimination	DNA fingerprinting to link cases; survival analysis	All secondary cases captured in surveillance data; genetic matching accurately reflects transmission patterns	Basic	Dutch index cases: 0.23, non-Dutch index cases: 0.25
Borgdorff [49]	San Francisco, USA, 1991–1996	RFLP data	Determine tuberculosis transmission dynamics in San Francisco and its association with country of birth and ethnicity	Define effective reproductive number as a function of transmission index, which is a function of number of secondary cases and potential source cases in a given subgroup	Each cluster originates from a single source case in the database; either the first case of a cluster was its source case, or that the probability of being a source case declined exponentially over time by 0.77% per day	Effective, recent transmission	0.24 (95% CI 0.17–0.31)
Davidow [50]	New York City, 1989–1993	TB and AIDS surveillance data	Evaluate the importance of recent M. tuberculosis transmission	Estimated # of TB infectious cases 1 year ago and computed short-term R0; R0 = the average # of new infections caused by each case per year of infectiousness*the average duration of infectiousness*the probability of progressing to active TB within 1 year after infection	Some clinical assumptions; parameter values in equation taken from the literature or calculated from neighbourhood-specific data	Short-term	No explicit estimates; focused on percentage of TB cases due to infection 1 year ago
Vynnycky [51]	England and Wales, 1900	Surveillance data; age and time-specific mortality rates	Describe transmission dynamics of all forms of pulmonary TB	Age-structured mathematical model with compartments for endogenous and exogenous diseases	General relationship between: first primary episode and age at infection, risk of exogenous disease and age at reinfection, endogenous disease and current age; risk of reinfection and first infection are identical; parameter values from the literature	Basic and net which is the same as effective	Net R at about 1 from 1900–1950; basic R0 declined from about 3 in 1900, reached 2 by 1950, and first fell below 1 in about 1960
Salpeter [52]	USA, 1930–1995	Case rates, correction for rates before 1975	Estimate time delay from infection to disease and R	Estimate R as a function of case rate and the shape of the delay function	R and case rate constant with calendar time t; incidence rate of latent infection is independent of the age	Effective	0.55, range 0.4–0.7 in sensitivity analysis
Borgdorff [27]	Netherlands, 1993–1995	RFLP data	Quantifying transmission of TB between and within nationalities	Effective R0 estimated as a function of transmission index	Probability of a patient being the source of a cluster was proportional to the incidence rate of potential sources times the probability that a potential source would give rise to a cluster	Effective	0.26, 95% CI (0.20–0.32); also estimated for different nationalities