Quantifying TB transmission: a systematic review of reproduction number and serial interval estimates for tuberculosis

Abstract

Tuberculosis (TB) is the leading global infectious cause of death. Understanding TB transmission is critical to creating policies and monitoring the disease with the end goal of TB elimination. To our knowledge, there has been no systematic review of key transmission parameters for TB. We carried out a systematic review of the published literature to identify studies estimating either of the two key TB transmission parameters: the serial interval (SI) and the reproductive number. We identified five publications that estimated the SI and 56 publications that estimated the reproductive number. The SI estimates from four studies were: 0.57, 1.42, 1.44 and 1.65 years; the fifth paper presented age-specific estimates ranging from 20 to 30 years (for infants <1 year old) to <5 years (for adults). The reproductive number estimates ranged from 0.24 in the Netherlands (during 1933–2007) to 4.3 in China in 2012. We found a limited number of publications and many high TB burden settings were not represented. Certain features of TB dynamics, such as slow transmission, complicated parameter estimation, require novel methods. Additional efforts to estimate these parameters for TB are needed so that we can monitor and evaluate interventions designed to achieve TB elimination.

Introduction

Tuberculosis (TB), an airborne bacterial infection caused by the organism Mycobacterium tuberculosis (Mtb), has surpassed HIV/AIDS as the leading cause of death due to a single infectious organism worldwide [1]. It primarily attacks the lungs but can also infect other areas of the body [2, 3]. Those exposed to Mtb often develop latent TB infection (LTBI) and have a 5–10% lifetime risk of progressing to active TB [4, 5]. Worldwide, 2–3 billion people are infected with TB; an estimated 10.4 million people developed active TB disease in 2015 [4]. Major innovations in strategies and tools to monitor the success of new strategies are needed to achieve the World Health Organisation (WHO)’s ENDTB goals of reducing TB deaths by 95% and new cases by 90% by 2035 [4].

The reproductive number and serial interval (SI) are two key quantities in describing transmission of an infectious disease. The reproductive number is defined as the average number of secondary cases a primary infectious case will produce. In a totally susceptible population, it is referred to as the basic reproductive number (R₀); it is referred to as the effective reproductive number (R_e) if the population includes both susceptible and non-susceptible persons [6]. An R_e > 1 indicates that the disease will continue to spread while an R_e < 1 indicates that the disease will eventually die out. Although the reproductive number is usually defined as the average number of secondary cases, it is occasionally defined as the average number of secondary infections [7–10], a distinction that is important for a disease with a long incubation period (the time between infection and developing symptomatic disease) and/or only a fraction of infections progressing to disease. Depending on the setting, the reproductive number can be expressed as a function of parameters such as infection rate, contact rate, recovery rate, making it useful in determining whether or not a disease can spread through a population.

The serial interval (SI), defined as the time between disease symptom onset of a case and that of its infector [11], is a surrogate for the generation interval – an unobservable quantity defined as the time between the infection of a case and the time of infection of its infector [12]. The SI is an important quantity in the interpretation of infectious disease surveillance data, in the identification of outbreaks and in the optimisation of quarantine and contact tracing.

These two quantities have been used to inform control policies during outbreaks [13] by quantifying the transmission of infectious diseases such as influenza A (H1N1) [11, 12, 14], Severe Acute Respiratory Syndrome (SARS) [12, 15] and Ebola [16, 17], where progression to disease upon transmission occurs quickly. For example, Wallinga and Teunis [18] in 2004 demonstrated the impact of the first global alert against SARS on the change of the effective reproductive number.

TB has a slower transmission rate due to its much longer incubation period. Of the 5–10% of infections that develop into active (symptomatic and infectious) TB disease, it is thought that the majority occur within the first 2 years after infection [2, 5, 19], although active TB disease can develop decades after initial infection [20]. This is much longer than the aforementioned infectious diseases where cases show symptoms within days of infection. Although there is an increasing consensus that some transmission events may occur before the infector shows symptoms, many likely occur after the infector is symptomatic, therefore, the longer the incubation period is, the longer the SI (Fig. 1).

Fig. 1.

Important infectious disease intervals. The time between a and c is the serial interval; the time between b and c is the incubation period.

Development of TB disease can be caused by de novo infection, reactivation of the same bacterial strain as a previous infection [5, 21] or by infection with a bacterial strain different from the original infection (reinfection TB). This complicates estimation of the serial interval, unless molecular techniques are used to distinguish reinfection and reactivation [21]. To our knowledge, there has been no systematic review of methods to estimate the serial interval and reproductive number for TB. Therefore, in this paper we systematically review the literature to examine the methods applied to the estimation of TB transmission parameters and the estimates obtained from these methods. This compilation informs the gaps in our understanding of TB and identifies areas where further research is needed to develop methods to better understand TB transmission.

Methods

We conducted two searches in PubMed for publications in English – one for TB and serial interval; one for TB and reproductive number.

Tuberculosis and serial interval

(‘Tuberculosis’[MeSH] OR ‘Mycobacterium tuberculosis’[MeSH] OR ‘tuberculosis’[TI]) and (‘serial interval’[tiab] or ‘generation interval‘[tiab] or ‘serial distribution’ [tiab] or ‘secondary infections’ [tiab] or ‘secondary cases’ [tiab]).

TB and reproductive number

(‘Tuberculosis’[MeSH] OR ‘Mycobacterium tuberculosis’[MeSH] OR ‘tuberculosis’[TI] OR ‘pulmonary, tuberculosis [MeSH]’) and (‘reproductive number’[tiab] or ‘reproduction number’[tiab] or ‘reproductive rate’[tiab] or ‘reproduction rate’[tiab] or ‘reproduction ratio’[tiab] or ‘reproductive ratio’[tiab] or ‘reproduction value’[tiab] or ‘reproductive value’[tiab] or ‘R₀’[tiab] or ‘secondary infections’[tiab] or ‘secondary cases’[tiab]).

Titles and abstracts of the publications referenced in the articles we found were reviewed for inclusion for either parameter. For the SI, as limited number of publications met our inclusion criteria, we also reviewed the titles and abstracts of publications that cited the serial interval articles that we included in a full-text review.

Two reviewers (two of YM, HEJ, LFW) independently screened all titles and abstracts, resolving discrepancies by consensus. Each publication was then independently reviewed by two reviewers (two of YM, HEJ, LFW) for inclusion. From the included articles, the same pairs of reviewers extracted the following details for all parameter estimates (if available): point estimates, confidence intervals, ranges, sample size and location/setting. We summarised the methods for analysis and aggregated those with similar estimation approaches.

Results

Serial interval

The serial interval query returned 171 articles (Fig. 2), of which 163 were excluded as they did not present any estimates. Leung et al. [22] reported the serial interval as the time from identification of primary case to secondary case as median 1.4 years (range: 0.4–5.2 years). This study used household transmission data from Hong Kong and focused on MDR- and XDR-TB. Vynnycky and Fine [23] analysed a population of white males in England and Wales in the 20th century using a mathematical compartmental model to estimate the SI as dependent on the age when infection occurred, distinguishing reinfection and reactivation in the model. In this model, the risk of developing disease was calibrated on incidence data. The estimates were presented as a frequency distribution. The most frequent time to develop disease was estimated at: between 20 and 30 years due to reinfection for those infected in the first year of life; between 10 and 14 years due to reinfection for those infected at age 10; <5 years due to recent infection for those infected at age 20 and those infected at age 40. ten Asbroek et al. [24] analysed genetic data for a Dutch sample from 1993 to 1996 to link infectors and infected people using DNA fingerprinting based on restriction fragment length polymorphism (RFLP) and estimated the serial interval at a geometric mean of 0.57 years (95% confidence interval (CI) 0.44–0.73). In this 4-year study, the probability of observing both the infector and the infected person depended on the time interval between isolates – the shorter this time interval was, the more likely that this couple was observed. Therefore, the observed serial intervals were weighted by the inverse of the difference between the length of the study period and the time between isolates of the infector and the infected person, allowing a rough correction for underrepresentation of longer SIs (Table 1).

Fig. 2.

Flow diagram of articles included in the search of estimates of the serial interval.

Table 1.

Estimates of the Serial Interval

Name of first author [publication]	Objective	Method	Assumptions	Estimated serial interval
Leung [22]	Study household transmission of MDR-TB	Data on all MDR-TB in Hong Kong 1997–2006. Did contact investigations and DNA fingerprinting and linked index to secondary cases	No censoring in this estimate, not clear how long people were followed up for	1.4 (0.4–5.2) years (median)
Vynnycky [23]	Demonstrate how the lifetime risk of disease, the incubation period and the serial interval changed	An age-dependent compartmental model	Assumed values for model input parameters such as the annual risk of infection	Estimated as dependent on age of infection and summarised as frequency distributions
ten Asbroek [24]	To determine the serial interval and incubation period of tuberculosis within 4 years of transmission	Descriptive approach on RFLP data (used to link infectors and infected people)	One source of infection for each infected cluster	0.57 years (95% CI 0.44–0.73)
Borgdorff [25]	Same as [24]	Same as [24]	NA	Median: 1.44 years (95% CI 1.29–1.63)
Brooks-Pollock [26]	Estimate the relative contributions of household and community transmission, the serial interval and the immunity afforded by a previous TB infection	Descriptive approach for the serial interval	All members of the study cohort have been exposed to TB by living with someone with active disease	Mean serial interval: 3.5 years; median serial interval 1.65 years

Two articles that cited the articles that met our inclusion criteria in the PubMed search reported estimates of the SI and were included for full-text review. Borgdorff et al. [25] used the same method on genetic data as [24] to estimate the median SI as 1.44 years (95% CI 1.29–1.63 years) for a Dutch sample from 1993–2007. Brooks-Pollock [26] in 2011 analysed cross-sectional household data for a sample in Lima, Peru from 1996 to 2002 and reported the time between the diagnosis of the infector and the infected person as an estimate for the SI with mean at 3.5 years and the median at 1.65 years.

Reproductive number

Two hundred and thirty-seven articles were identified for the reproductive number of TB. Additionally, six articles were included based on reviewing titles and abstracts of the articles that were referenced in the 237 articles, making the total number of articles 243. Fifty-six articles met our inclusion criteria and are described below. Three articles used either approximate Bayesian or exact likelihood methods, 24 articles used either a mathematical model fit with empirical data or a descriptive/regression approach on empirical data, and 29 articles used a simulation-based mathematical model (Fig. 3). Explicit estimates were extracted and summarised in Fig. 4. The estimates range from as low as 0.26 for the Netherlands in 1993–2007 to as high as 4.3 in China in 2012.

Fig. 3.

Flow diagram of articles included in the search of estimates of the reproductive number.

Fig. 4.

Reproductive number from studies with explicit R estimate from empirical data. Notes: (1) The range is for years 2005-2012, with the reproductive number estimated at 3.33, 3.72, 3.38, 3.97, 4.29, 3.32, 3.92 and 4.30, respectively. (2) For each location, the first R corresponds to drug-sensitive population and the second correspond to drug-resistant population. (3) R estimated for 35 states and union territories of India with estimates ranging from 0.72 to 0.98; 0.92 is the overall estimate for India. (4) For each location, the first R corresponds to drug-sensitive population and the second correspond to drug-resistant population. (5) Bordgorff in [27–29] estimated the reproductive number for the Netherlands from 1993 to 2007 at around 0.26 with lower bound of the 95% CI around 0.20 and upper bound around 0.32. (6) Broken lines indicate range; solid lines indicate 95% confidence interval. (7) Vynnycky and Fine [23] in 1998 estimated the basic reproductive number to decline from about 3 in 1900 to 2 in 1950 and to below 1 in about 1960 for England and Wales, which is not included in this graph.

Three articles (Table 2) used the same genetic RFLP data from TB diseased individuals during an outbreak in San Francisco in 1991–1992 [33]. They all estimated the effective reproductive number in a Bayesian framework. Tanaka et al. [30] used an approximated computation method to obtain an estimate of 3.4 (95% CI 1.4–79.7). Stadler [31] in 2013 used an exact likelihood method to obtain an estimate of 1.02 (95% CI 1.01–1.04) and claimed that the difference from the estimate in [30] was due to the lack of precision in the approximation of the posterior distribution in [30]. Aandahl et al. [32] in 2014 reconciled the two methods by specifying an informative prior for two parameters in [30] and improving the convergence performance of the Markov chain Monte Carlo (MCMC) sampler in [31]. The reconciled estimates were: 2.1 (95% CI 1.54–2.66) for the approximate method in [30] and 2.05 (95% CI 1.55–2.63) for the exact method in [31]. These papers used the same model but differed in the methods used to obtain the estimates. The assumptions of the model are listed in Table 2.

Table 2.

Estimates of the reproduction number using approximate Bayesian computation and exact likelihood methods (all methods used data from San Francisco on cases reported in 1994)

Name of first author [publication]	Objective	Method	Assumptions	Estimated reproductive number (95% credible interval)
Tanaka [30]	Estimate TB transmission parameters: net transmission rate, doubling time and reproductive number	Approximate Bayesian computation	Constant supply of susceptible people; all genotypes are selectively neutral; mutation and transmission are independent; infinite alleles; epidemic spreads until N individuals are infected	3.4 (1.4–79.7)
Stadler [31]	Estimate TB transmission parameters: net transmission rate and reproductive number	Exact likelihood	Constant birth-death rate; infinite alleles; epidemic started at a random time in the past; an isolate is sampled from an individual with probability P	1.02 (1.01–1.04)
Aandahl [32]	Reconcile the different estimates in [30, 31]	Improved the method in [30] by specifying informative priors; improved convergence performance of the MCMC sampler in [31]	Fixed mutation rate; used Gaussian prior for the death/recovery rate	2.1 (1.54–2.66) in [30] 2.05 (1.55–2.63) in [31]

Twenty-four articles analysed the reproductive number with empirical data (Table 3). Seventeen articles reported explicit estimates, with five estimating the effective reproductive number and 12 estimating the basic reproductive number. The majority of these articles used mathematical compartmental models with different variations in structure and parameterisation to address issues such as seasonality [43], the effect of age [46, 51] and HIV–TB co-epidemics [9].

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

Two articles [7, 40] used the Wells–Riley model or a modified version of the model. In these models, the reproductive number was expressed as a function of infection risk, which was further expressed as proportionate to environmental factors such as the number of infectious people in a given space, per-person breathing rate and inversely proportionate to germ-free ventilation rate. One article derived the reproductive number as a function of the transmission index – defined as the ratio of the number of secondary cases to the sum of the number of source cases (infectors) and non-clustered cases where clusters are defined as groups of patients that had isolates with identical fingerprints [27]. The largest reproductive number (effective) was estimated in [38] using the Chinese Centre for Disease Control and Prevention (CDC) data from 2005 to 2012, where the annual reproductive number ranged between 3.33 and 4.30 for years 2005–2012 in China. The lowest (effective) reproductive number was estimated at 0.24 (95% CI 0.17–0.31) using RFLP data in San Francisco, USA from 1991 to 1996 [49]. Vynnycky and Fine [51] in 1998 used an age-structured mathematical model and estimated the effective reproductive number to be around 1 from 1900 to 1950 in England and Wales; the basic reproductive number was estimated to have declined from about 3 in 1900 to 2 by 1950, and first fell below 1 in about 1960. The assumptions of these models are listed in Table 3.

One article defined the reproductive number as the number of secondary infections caused by an infectious case [7]. As only a fraction of the infected people develops active disease, the estimated reproductive number was larger than those in the other papers. The median of the reproductive number in this article ranged from 14 to 45 as exposure time increased from 1 to 5 months.

Twenty-nine articles analysed the reproductive number through simulation based on a mathematical modelling framework (Table 4). These articles all used mathematical compartmental models with different variations to address issues such as reinfection [68], the interaction between HIV and TB [64], and drug-resistant and drug-sensitive TB [60]. The majority of them focused on studying the effect of these issues on TB transmission dynamics through simulations that were not based on a specific population. In this case, parameters for the model were based on estimates from studies performed in diverse settings or sampled over a range of feasible values. The analytical expression of the basic reproductive number was derived to study the disease-free equilibrium and endemic-persistent state of TB in these papers. Five articles [10, 60, 64, 73, 76] included drug-resistant TB cases as a compartment and four articles [58, 68, 72, 75] included HIV + TB cases as a compartment.

Table 4.

Estimates of the reproductive number from mathematical models based on simulation

Name of first author [publication]	Objective	Methods, setting	Assumptions	Basic or effective R0	Estimated R0
Ren [53]	Develop SEIR model for imperfect treatment with age-dependent latency and relapse	SEIR model	TB infectious in latent period; age-dependence	Basic	Dependent on parameters
Jabbari [54]	To set up a model that can examine two TB strains (DS and DR) with multiple latent stages	Mathematical compartmental model with compartments for latency stages	The drug-sensitive strain will not play a role in the process of exogenous reinfection for the drug-resistant strain	Basic	Dependent on parameters
Okuonghae [55]	Study the effects of additional heterogeneities from the level of TB awareness on TB transmission dynamics and case detection rate	Expanding [34] by dividing both susceptible and latently infected compartment by level of TB awareness	Reasonable values and bounds for parameters such as transmission rate, recovery rate from the literature	Effective	Dependent on parameters such as active case finding rate and treatment rate
Liu [56]	Evaluate effect of treatment for TB	Compartmental model with treatment and two latent periods incorporated	Once the treatment of active TB cases is interrupted, there is no more treatment; specified model parameter values and their relationship with one another	Basic	Dependent on transmission coefficients
Silva [57]	Study optimal strategies for the controlling active TB infectious and persistent latent individuals	Compartmental model considering reinfection and post-exposure interventions with the addition of early latent and persistent latent compartments	Parameter values taken from the literature	Basic	Dependent on transmission coefficient
Hu [58]	Study the threshold dynamics of TB	Compartmental model with periodic functions for reactivation rate and infection rate; include additional compartment for treated people that do not return to the hospital for examination	NA	Basic	Dependent on transmission coefficient
Emvudu [59]	Address the problem of optimal control for TB transmission dynamics	Compartmental model with an additional compartment for loss to follow-up	Half of the parameter values were assumed; others taken from the Cameroon literature	Basic	Dependent on parameters such as transmission rate
Sergeev [60]	How drug-sensitive and drug-resistant strains mixed together can impacts long-term TB dynamics	Compartmental with the three compartments for both latent and infected: drug-resistant, drug-sensitive and mixed strains	Reasonable values for many parameters; few data exist to inform model parameters	Basic; estimated for drug-resistant, drug-sensitive and mixed strains	Dependent on model parameters
Roeger [61]	Model TB and HIV co-infection	Compartmental model for joint dynamics of TB and HIV and compute independent reproductive numbers for the two diseases	Probability of infection is the same for those treated with TB and those susceptible; assumed relationship among model parameters	Overall R0 as the max of R0 for TB and HIV	Dependent on model parameters
Gerberry [62]	Study the trade-off between BCG and detection, treatment of TB	Compartmental model with additional compartments for latently infected and unvaccinated, latently infected and vaccinated; establish thresholds for basic R0	Throughout the duration of the vaccine's efficacy, latent TB completely undetectable	Basic	Dependent on model parameters
Bhunu [63]	Model HIV/AIDS and TB coinfection	Compartmental model for TB, HIV separately without intervention; full model with intervention	Parameter values from Central Statistics Office of Zimbabwe and literature; relationship amongst parameters in the model	Basic	Dependent on model parameters
Bhunu [8]	Model the effect of pre-exposure and post-exposure vaccines	Compartmental model with additional compartments for susceptible (vaccinated or not) and latent (history of vaccine or not)	Homogeneous mixing; recovered people would not develop disease from reinfection, but could be re-infected; parameter values taken from Central Statistics Office and literature	Basic	Dependent on model parameters
Sharomi [64]	Address the interaction between HIV and TB	TB-only, HIV-only and full model analysed with both susceptible and latent compartments divided according to TB and HIV status	Dually infected people could not transmit both diseases; some parameters taken from the literature, others assumed	Basic	Dependent on model parameters
McCluskey [65]	Address global stability of high dimensional TB model	Use Lyapunov function to demonstrate the stability of the endemic equilibria in mathematical models for TB: SEIR, SEIS and SIR; fast and slow progression incorporated		Basic	No explicit estimate
Martcheva [66]	Address the issue of an infected person being subject to further contacts with infectious individuals—‘super infection’	Subdivide the latent stage into one where the disease progresses and one where the disease development is on hold	Relationship among model parameters	Basic	No explicit estimate
Aparicio [67]	Express basic R0 as a function of cluster size	Divide individuals into either active clusters or otherwise	Homogeneous mixing	Basic	No explicit estimate; expressed as a function of household size
Feng [68]	Examine how exogenous reinfection changes the TB transmission dynamics	Include additional parameters in the mathematical model to model exogenous reinfection	Constant per capita removal rate to focus on the role of reinfection	Basic	No explicit estimate; analytical expression
Beatriz [69]	Assess the effects of heterogeneous infectivity	Divide infective period into k stages	Homogenous mixing; bilinear incidence rate	Basic	No explicit estimate; analytical expression
Castillo-Chavez [70]	Use an age-structure model to study the dynamics of TB	Use age-specific parameters in the compartmental model; transmission dynamics studied for with and without vaccine	Mixing between individuals is proportional to their age-dependent activity level; disease-induced death rate neglected	Net and basic	No explicit estimate; analytical expression
Lietman [71]	Test the hypothesis that exposure to TB leads to disappearance of leprosy	Add in leprosy compartment in the mathematical model	Cross-immunity is symmetric: same immunity for TB and leprosy	Basic	Dependent on R0 of leprosy and cross-protection rate
Sanchez [72]	Evaluate the effects of parameter estimation uncertainty on the value of R0	Latin hypercube sampling used on parameters in the compartmental model in Blower [72] to evaluate uncertainty of R0	Range for parameters in the compartmental model	Sum of R0 for fast, slow and relapse	Dependent on parameters in the model
Gumel [10]	Study the transmission dynamics of TB with multiple strains, in the presence of exogenous reinfection	Included drug-sensitive and resistant strains in the compartmental model; exogenous reinfection incorporated	Homogenous mixing	Effective R0 for the two strains	Dependent on parameters in the model
Singer [73]	Study the impact of different reinfection levels of latently infected individuals on TB transmission dynamics	Compartmental model for heterogeneous population: one group more susceptible to infection than the other	Parameter range uniformly distributed according to previous papers	Basic	No explicit estimate
Trauer [74]	Model TB transmission for highly endemic regions of the Asia-Pacific where HIV-coinfection is low	Compartmental models with compartments for immunisation, latency, reinfection, drug-resistance, etc.	Parameters fixed values according to papers and WHO	Basic	Dependent on parameters; computed as 8.34 for drug-susceptible and 5.84 for drug resistant at baseline
Dye [75]	To establish criteria for MDR-TB control	Compartmental models with compartments for drug-susceptible, drug-resistant, treatment failure, etc.	Parameters calculated from different populations	Basic	Dependent on parameters; best estimated of the model parameters yielded R0 = 1.6 (95% CI 1.02–2.67)
Blower [76]	Track the emergence and evolution of multiple strains of drug-resistant TB	Non-compartmental mathematical model	NA	Basic	Dependent on drug susceptibility of TB
Blower [77]	Model the transmission dynamics of TB	Compartmental models with latently infected, infectious, non-infectious, recovered compartments	Some model parameters assumed; some taken from references	Basic; defined as the sum of slow progression, recent transmission and relapse	Median of 4.47, range: 0.74–18.58
Blower [78]	Understand, predict and control TB	Compartmental models with drug-sensitive and drug-resistant compartments	NA	Basic	Dependent on model parameters
Aparicio [67]	Evaluate homogeneous mixing and heterogeneous mixing models for TB	Three types of compartmental models: a standard incidence homogenous mixing mode; a heterogeneous mixing model; an age-structured model	Assumptions on model parameters	Basic	Dependent on model parameters

Discussion

We found very few publications that reported estimates for the serial interval of TB. Estimates of the reproductive number were limited to seven countries, with the majority of the publications using mathematical compartmental models that did not base estimates on actual data. This indicates a need for a better understanding of these crucial parameters of TB transmission, which can help inform public health decisions in order to reach the WHO's End TB goals [4] of reducing TB deaths by 95% and incident cases by 90% by 2035.

Serial interval

We found only five articles that discussed the estimation of the SI for TB and presented explicit estimates. ten Asbroek [24] estimated the serial interval over 4 years as a geometric mean of 0.57 years (95% CI 0.44–0.73). Using the same method over a longer study period (15 years compared with 4 years in [24]), the estimated median was 1.44 years, which is comparable with the median serial interval of 1.65 years in [26] with a 6-year study period. This indicates that the study period could potentially bias the serial interval estimates, even though the method in [24] corrected for the underrepresentation of longer serial intervals. In contrast with other infectious diseases that progress much faster and have SIs measured in days, the SI of TB can be weeks, years and even decades [23]. This unique feature of TB makes it difficult to obtain an unbiased estimate of the SI as lengthy follow-up is required to observe the long period between presence of symptoms of the infector and the infected person. Additionally, uncertainty regarding the presence and impact of multiple infection events further complicates the observation of this interval. Currently, the most common way of monitoring TB is by looking at annual incidence rates in studies that are often no longer than 5 years [79, 80]. This creates two issues: right censoring as symptoms of the infected people can develop long after the end of studies, and interval censoring as the symptom onset time can fall during long intervals between two observed time points. Another issue is patients’ and doctors’ delay. Patients may not seek medical assistance immediately after symptoms develop and diagnosis may require lab-processing time which causes delay in establishing the diagnosis [24], creating a left censoring issue. Survival analysis techniques can be considered to address these issues but may need substantial modification. Further ambiguity exists due to the inconsistent availability of genetic typing of strains to link cases, and the further uncertainty about how to best link strains when genetic information is available, as such information may not account for mutation rate, or infection with multiple bacterial strains.

Reproductive number

The majority of the articles used mathematical compartmental models (a brief introduction can be found in the appendix) to describe the transmission dynamics of TB. These models have been widely used to understand the dynamics of infectious diseases including SARS, influenza and TB, and they either use empirical data to estimate the parameters in the model or are based on simulation.

The compartmental models using empirical data are distinguished from simulation-based models in two key ways. First, empirical models use data to estimate some of the model parameters, while others are taken directly from the literature or assumed. Simulation-based models do not use empirical data to parameterise the models. For example, in [42] where empirical data was used, the mortality rate due to drug susceptible TB was estimated from Taiwanese Centre of Disease Control data and the effective contact rate for TB was estimated based on the literature; in [41] where simulation was used, the recruitment rate was taken from the literature and awareness rate of TB was estimated from data.

A second distinction between models based on empirical data and simulation-based models is that the former often report explicit estimates of the reproductive number for a specific region, while the latter usually focus on studying the impact of a certain feature on TB transmission dynamics. For example, in [37] where empirical data were used, the reproductive number was reported for India overall and by regions; in [60] where a simulation-based approach was used, the impact of drug-sensitive and drug-susceptible strains mixed together on TB transmission dynamics was studied.

In developed countries, the reproductive number was sometimes estimated to be well below 1: for example, 0.55 in the USA from 1930 to 1995 [52] and 0.26 in the Netherlands from 1993 to 1995 [27]. In developing countries, the reproductive number was as high as 4.3 in China in 2012 [38] and 3.55 in Southern India from 2004 to 2006. In the Netherlands, the reproductive number has been consistently estimated at well below one, ranging from 0.24 [49] to 0.48 [39].

The same dataset in San Francisco, USA in 1991–1992 (published in 1994) was used to estimate the effective reproductive number in two separate studies [30, 31] that yielded disparate results. The estimates from these two papers were reconciled in [32] to an estimated effective reproductive number of approximately 2.1 by specifying an informative prior for two parameters in [30] and improving the convergence performance of the MCMC sampler in [31]. One can contrast this estimate with other estimates for the USA to see the range of values obtained. A study of the entire USA in [52] estimated the reproductive number to be 0.55 using case rates of active TB in USA from 1955 to 1994. As shown in [81], TB incidence in San Francisco peaked between 1991 and 1993, due to the TB/HIV co-epidemic, which is consistent with the higher estimated reproductive number (around 2.1) in [30–32]. When using TB case rates in the entire USA from 1955 to 1994 as in [52], the potential geographical and temporal heterogeneity in the estimates is not well represented, resulting in an estimated reproductive number of 0.55. We would expect a lower reproductive number, and in particular, a reproductive number below one, when using data from 1955 to 1994 because by 1955, effective antibiotics were in use and BCG had also been developed, both leading to a reduction in TB incidence across the USA. In addition, Borgdorff [49] reported an effective reproductive number of 0.24 using RFLP data in San Francisco from 1991 to 1996. In this paper, the ratio of secondary cases and source cases was used to estimate the reproductive number, which may be an oversimplified estimator of the reproductive number. Issues such as linking the secondary cases and the sources cases have not been addressed. These divergent results indicate the need for the use of whole genome sequencing (WGS), which can be used to effectively link source and secondary cases.

Similar to the more statistical analysis of the San Francisco and the entire USA data, we observe that mathematical models lead to inconsistent results, at least partially attributable to the varying assumptions they make in their structure and parameterisation. For example, even though both [38] and [42] used mathematical compartmental models with different variations for similar regions (China and Taiwan), they have quite different estimates: between 3.3 and 4.3 in China from 2005 to 2012 as compared with 0.9 for drug-sensitive TB, around 0.38 for multidrug-resistant TB (defined as a TB strain resistant to at least isoniazid and rifampicin) in Taiwan from 2005 to 2010. Both articles used incidence data from Chinese and Taiwanese CDC but formulated the compartments in the models differently. In [38], compartments ‘exposed’, ‘infectious and hospitalised’ and ‘infectious but not hospitalised’ were included; in [42], compartments ‘latent’, ‘infected’ were used for two sub-populations: drug-sensitive and multidrug-resistant. The model parameters were also differently specified: in [38], some parameters were assumed while others were estimated using minimum sum of square; in [42], some parameters were given a probabilistic distribution and estimated with a root-mean-squared error method while others were assumed. The difference between the estimated reproductive numbers produced from these two modelling exercises is striking, as the two regions and populations are quite comparable in terms of demographics, economic status and access to healthcare. One could similarly contrast the modelling approaches and estimates obtained in [34] and [43], two other studies from China and Taiwan from similar time periods that produced different estimates. The differing model structures, as well as the parameter estimates, including the recruitment rate, incidence rate, and mortality rate, likely drive these observed differences. It is difficult to say which model might be a more accurate reflection of reality.

The example above illustrates the challenges of interpreting and using mathematical models for estimation of the reproductive number. However, most estimates to date make use of this approach. One shortcoming of these models is that they require assumptions about parameter values that may be difficult to estimate, such as the transmission rate, the treatment rate and the recovery rate, which are often unobservable and not reliably estimated. As a result, most of the articles assume values for the parameters in the model based on evidence in the published literature, where it exists, sometimes without measures of uncertainty (e.g. standard errors). Model structure also varies substantially from study to study, with no generally agreed upon approach to model TB and estimate parameters. For example, in [35], compartments of different vaccine strategies were included in the model and in [38], a compartment of hospitalisation was included in the model. These models also often require assumptions about the parameters used to run the models, which are likely to differ by country and time period. Sometimes sufficient data are unavailable to parameterise a model and generalisations need to be made that may not always be appropriate. The majority of the existing publications use mathematical compartmental models, which are not often ideal for statistical inference and estimation due to strong assumptions for the model structure and parameters used to run the models. While these models have the flexibility of using different compartments to evaluate the impact of policies, they are not ideal for real-time analysis where the appropriate model structure and parameter values required fitting the model may not be clear. The complexity of the natural history of TB and important factors such as HIV and drug resistance complicate these models and require additional parameters for which the data are sometimes not available. We believe that it is important to develop, as a complementary approach to compartmental models, likelihood-based data-driven analytic tools. Ideally, these estimators can be used with datasets using minimal assumptions. In addition, as WGS data become more ubiquitous [82], it will be important to develop methods that use these data to estimate the reproductive number (Fig. 5).

Fig. 5.

Shaded areas and stars indicate countries and cities with reproductive number estimates. Multiple estimates: China, Taiwan, USA, India; one estimate: Ukraine, the Netherlands, South Africa, the UK. *indicates San Francisco corresponding to data used in [30–32].

This review found that the reproductive number estimates for TB are very divergent – in reality, we would expect different results in different parts of the world, reflecting diversity in TB epidemics geographically. Therefore, it is important to have estimates from a wide range of settings. An ultimate goal of methods to estimate the reproductive number should be to use routinely collected data (including potentially WGS data) to be able to monitor the reproductive number in ‘real-time’ and evaluate interventions through this process.

Our review is subject to a number of limitations. It is possible that some useful papers could have been excluded due to our selection of search terms and our inclusion of reports in only English. These limitations are difficult to avoid in systematic reviews, in which the potential for increased yield from a wider search must be weighed against the increased feasibility of a tighter search. Additionally, our query was limited to searching in abstracts and titles, making it possible that we excluded articles where the keywords only appear in the text [25].

In conclusion, a limited number of studies have yielded explicit estimates for the serial interval and reproductive number of TB. When estimating the serial interval, it is difficult to observe the symptom onset of the infector and infected person with precision. Estimates of the reproductive number were limited geographically (Fig. 6) with estimates only available for seven countries. Settings with high TB burdens, especially high drug-resistant TB burdens such as the former Soviet Union [83] are not included in these papers. In addition, there was only one estimate from a high TB and high HIV burden country [47]. The lack of estimates could be because incidence and mortality rates are currently used to monitor TB control. These rates are not suitable for monitoring transmission; reductions in mortality could be attributed to improvements in treatment outcomes rather than any change in transmission and, due to the long incubation period of TB, changes in transmission could take years to impact incidence rates. In contrast, the reproductive number can provide a direct estimate of TB transmission itself. Most studies used mathematical models with various assumed model structures and parameters, making it difficult to compare the estimates and draw useful conclusions about the TB transmission dynamics by evaluating the reproductive number.

Fig. 6.

Examples of mathematical compartmental models.

The WHO End TB goals [4] include reducing TB deaths by 95% and incident cases by 90% by 2035. To achieve these goals, it is necessary to obtain improved estimates of the reproductive number and the SI as they can be used for monitoring and evaluating the effect of interventions on TB transmission. For example, the serial interval of TB can be used to determine how long one must monitor contacts of an infectious TB case to see if they will develop symptoms [84]. The effective reproductive number can be used to monitor the efficacy of interventions in reducing transmission. As interventions decrease transmission, estimates of the reproductive number should correspondingly decrease [41]; in particular, if the reproductive number can be maintained below one, the disease can potentially be eliminated.

The limited number of articles that we found and the lack of geographic representation, demonstrate a substantial gap in our understanding of these crucial parameters of TB transmission in diverse settings.

Ethical standards

The authors assert that all procedures contributing to this work comply with the ethical standards of the relevant national and institutional committees on human experimentation and with the Helsinki Declaration of 1975, as revised in 2008.

Conflict of interest

None.