Mathematical Models of Tuberculosis Reactivation and Relapse

Structural and practical identifiability analysis in bioengineering: a beginner's guide

Advancements in digital technology have brought modelling to the forefront in many disciplines from healthcare to architecture. Mathematical models, often represented using parametrised sets of ordinary differential equations, can be used to characterise different processes. To infer possible estimates for the unknown parameters, these models are usually calibrated using associated experimental data. Structural and practical identifiability analyses are a key component that should be assessed prior to parameter estimation. This is because identifiability analyses can provide insights as to whether or not a parameter can take on single, multiple, or even infinitely or countably many values which will ultimately have an impact on the reliability of the parameter estimates. Also, identifiability analyses can help to determine whether the data collected are sufficient or of good enough quality to truly estimate the parameters or if more data or even reparameterization of the model is necessary to proceed with the parameter estimation process. Thus, such analyses also provide an important role in terms of model design (structural identifiability analysis) and the collection of experimental data (practical identifiability analysis). Despite the popularity of using data to estimate the values of unknown parameters, structural and practical identifiability analyses of these models are often overlooked. Possible reasons for non-consideration of application of such analyses may be lack of awareness, accessibility, and usability issues, especially for more complicated models and methods of analysis. The aim of this study is to introduce and perform both structural and practical identifiability analyses in an accessible and informative manner via application to well established and commonly accepted bioengineering models. This will help to improve awareness of the importance of this stage of the modelling process and provide bioengineering researchers with an understanding of how to utilise the insights gained from such analyses in future model development.

IFN-γ independent markers of Mycobacterium tuberculosis exposure among male South African gold miners

BACKGROUND

The prevalence of tuberculosis among men who work in the gold mines of South Africa is among the highest in the world, but a fraction of miners demonstrate consistently negative results upon tuberculin skin test (TST) and IFN-γ release assay (IGRA). We hypothesized that these "resisters" (RSTRs) may display unconventional immune signatures of exposure to M. tuberculosis (M.tb).

METHODS

In a cohort of RSTRs and matched controls with latent TB infection (LTBI), we profiled the functional breadth of M.tb antigen-specific T cell and antibody responses using multi-parameter flow cytometry and systems serology, respectively.

FINDINGS

RSTRs and LTBI controls both exhibited IFN-γ independent T-cell and IgG antibody responses to M.tb-specific antigens ESAT-6 and CFP-10. Antigen-specific antibody Fc galactosylation and sialylation were higher among RSTRs. In a combined T-cell and antibody analysis, M.tb lysate-stimulated TNF secretion by T cells correlated positively with levels of purified protein derivative-specific IgG. A multivariate model of the combined data was able to differentiate RSTR and LTBI subjects.

INTERPRETATION

IFN-γ independent immune signatures of exposure to M.tb, which are not detected by approved clinical diagnostics, are readily detectable in an occupational cohort uniquely characterized by intense and long-term infection pressure. Further, TNF may mediate a coordinated response between M.tb-specific T-cells and B-cells.

FUNDING

This work was supported by the US National Institutes of Health (R01-AI124348 to Boom, Stein, and Hawn; R01-AI125189 and R01-AI146072 to Seshadri; and 75N93019C00071 to Fortune, Alter, Seshadri, and Boom), the Doris Duke Charitable Foundation (Davies), the Bill & Melinda Gates Foundation (OPP1151836 and OPP1109001 to Hawn; and OPP1151840 to Alter), Mass Life Science Foundation (Fortune), and Good Ventures Fund (Fortune).

A noninteger order SEITR dynamical model for TB

This research paper designs the noninteger order SEITR dynamical model in the Caputo sense for tuberculosis. The authors of the article have classified the infection compartment into four different compartments such as newly infected unrecognized individuals, diagnosed patients, highly infected patients, and patients with delays in treatment which provide better detail of the TB infection dynamic. We estimate the model parameters using the least square curve fitting and demonstrate that the proposed model provides a good fit to tuberculosis confirmed cases of India from the year 2000 to 2020. Further, we compute the basic reproduction number as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Re _{0} \approx 1.73$\end{document}ℜ0≈1.73 of the model using the next-generation matrix method and the model equilibria. The existence and uniqueness of the approximate solution for the SEITR model is validated using the generalized Adams–Bashforth–Moulton method. The graphical representation of the fractional order model is given to validate the result using the numerical simulation. We conclude that the fractional order model is more realistic than the classical integer order model and provide more detailed information about the real data of the TB disease dynamics.

Resistance to Mycobacterium tuberculosis infection among highly TB exposed South African gold miners

BACKGROUND

Despite high exposure to Mycobacterium tuberculosis, a small proportion of South African goldminers resist TB infection. We determined, among long-service gold miners i) the proportion who were TB uninfected and ii) epidemiological factors associated with being uninfected.

METHODS

We enrolled HIV-negative gold miners aged 33-60 years with ≥15 years' service and no history of TB or silicosis. Miners were defined as TB uninfected if i) QuantiFERON-TB Gold Plus (QFT-Plus) negative or ii) in a stricter definition, QFT-Plus-negative and zero-response on TST and as resisters if they were of Black/African ethnicity and negative on both tests. Logistic regression was used to identify epidemiological factors associated with being TB uninfected.

RESULTS

Of 307 participants with a QFT-Plus result, median age was 48 years (interquartile range [IQR] 44-53), median time working underground was 24 years (IQR 18-28), 303 (99%) were male and 91 (30%) were QFT-Plus-negative. The odds of being TB uninfected was 52% lower for unskilled workers (adjusted odds ratio [aOR] 0.48; 95% confidence interval [CI] 0.27-0.85; p = 0.013). Among 281 participants of Black/African ethnicity, 71 (25%) were QFT-Plus negative. Miners with a BMI ≥30 were less likely to be TB uninfected (OR 0.38; 95% CI 0.18-0.80). Using the stricter definition, 44.3% (136/307) of all miners were classified as either TB uninfected (35; 26%) or infected, (101; 74%) and the associations remained similar. Among Black/African miners; 123 were classified as either TB uninfected (23; 19%) or infected (100; 81%) using the stricter definition. No epidemiological factors for being TB uninfected were identified.

CONCLUSIONS

Despite high cumulative exposure, a small proportion of miners appear to be resistant to TB infection and are without distinguishing epidemiological characteristics.

An efficient numerical simulation and mathematical modeling for the prevention of tuberculosis

The main purpose of this research is to use a fractional-mathematical model including Atangana–Baleanu derivatives to explore the clinical associations and dynamical behavior of the tuberculosis. Herein, we used a lately introduced fractional operator having Mittag-Leffler kernel. The existence and inimitability problems to the relevant model were examined through the fixed-point theory. To verify the significance of the arbitrary fractional-order derivative, numerical outcomes were explored from the biological and mathematical viewpoints using the values of model parameters. The graphical simulations show the comparison of the predictor–corrector method (PCM) and Caputo method (CM) for different fractional orders and the results indicated the significant preference of PCM over CM.

Phenotype Definition for "Resisters" to Mycobacterium tuberculosis Infection in the Literature-A Review and Recommendations

Tuberculosis (TB) remains a worldwide problem. Despite the high disease rate, not all who are infected with Mycobacterium Tuberculosis (Mtb) develop disease. Interferon-γ (IFN-γ) specific T cell immune assays such as Quantiferon and Elispot, as well as a skin hypersensitivity test, known as a tuberculin skin test, are widely used to infer infection. These assays measure immune conversion in response to Mtb. Some individuals measure persistently negative to immune conversion, despite high and prolonged exposure to Mtb. Increasing interest into this phenotype has led to multiple publications describing various aspects of these responses. However, there is a lack of a unified "resister" definition. A universal definition will improve cross study data comparisons and assist with future study design and planning. We review the current literature describing this phenotype and make recommendations for future studies.

Immunological mechanisms of human resistance to persistent Mycobacterium tuberculosis infection

Mycobacterium tuberculosis is a leading cause of mortality worldwide and establishes a long-lived latent infection in a substantial proportion of the human population. Multiple lines of evidence suggest that some individuals are resistant to latent M. tuberculosis infection despite long-term and intense exposure, and we term these individuals 'resisters'. In this Review, we discuss the epidemiological and genetic data that support the existence of resisters and propose criteria to optimally define and characterize the resister phenotype. We review recent insights into the immune mechanisms of M. tuberculosis clearance, including responses mediated by macrophages, T cells and B cells. Understanding the cellular mechanisms that underlie resistance to M. tuberculosis infection may reveal immune correlates of protection that could be utilized for improved diagnostics, vaccine development and novel host-directed therapeutic strategies.

IFN-γ-independent immune markers of Mycobacterium tuberculosis exposure

Exposure to Mycobacterium tuberculosis (Mtb) results in heterogeneous clinical outcomes including primary progressive tuberculosis and latent Mtb infection (LTBI). Mtb infection is identified using the tuberculin skin test and interferon-γ (IFN-γ) release assay IGRA, and a positive result may prompt chemoprophylaxis to prevent progression to tuberculosis. In the present study, we report on a cohort of Ugandan individuals who were household contacts of patients with TB. These individuals were highly exposed to Mtb but tested negative by IFN-γ release assay and tuberculin skin test, ‘resisting’ development of classic LTBI. We show that ‘resisters’ possess IgM, class-switched IgG antibody responses and non-IFN-γ T cell responses to the Mtb-specific proteins ESAT6 and CFP10, immunologic evidence of exposure to Mtb. Compared to subjects with classic LTBI, ‘resisters’ display enhanced antibody avidity and distinct Mtb-specific IgG Fc profiles. These data reveal a distinctive adaptive immune profile among Mtb-exposed subjects, supporting an expanded definition of the host response to Mtb exposure, with implications for public health and the design of clinical trials.

New immune biomarkers of exposure to tuberculosis may require a rethink of evidence of Mycobacterium tuberculosis infection and control.

Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains

The objective of this paper is to study the optimal control problem for the fractional tuberculosis (TB) infection model including the impact of diabetes and resistant strains. The governed model consists of 14 fractional-order (FO) equations. Four control variables are presented to minimize the cost of interventions. The fractional derivative is defined in the Atangana-Baleanu-Caputo (ABC) sense. New numerical schemes for simulating a FO optimal system with Mittag-Leffler kernels are presented. These schemes are based on the fundamental theorem of fractional calculus and Lagrange polynomial interpolation. We introduce a simple modification of the step size in the two-step Lagrange polynomial interpolation to obtain stability in a larger region. Moreover, necessary and sufficient conditions for the control problem are considered. Some numerical simulations are given to validate the theoretical results.