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 |