Mathematical Modelling of the Epidemiology of Tuberculosis

Modelling the preventive treatment under media impact on tuberculosis: A comparison in four regions of China

Preventive treatment for people with latent Tuberculosis infection (LTBI) has aroused our great interest. In this paper, we propose and analyze a novel mathematical model of TB considering preventive treatment with media impact. The basic reproduction number R 0 is defined by the next generation matrix method. In the case without media impact, we prove that the disease-free equilibrium is globally asymptotically stable (unstable) if R 0 < 1 ( R 0 > 1 ) . Furthermore, we obtain that a unique endemic equilibrium exists when R 0 > 1 , which is globally asymptotically stable in the case of permanent immunity and no media impact. We fit the model to the newly reported TB cases data from 2009 to 2019 of four regions in China and estimate the parameters. And we estimated R 0 = 0.5013 < 1 in Hubei indicating that TB in Hubei will be eliminated in the future. However, the estimated R 0 = 1.015 > 1 in Henan, R 0 = 1.282 > 1 in Jiangxi and R 0 = 1.930 > 1 in Xinjiang imply that TB will continue to persist in these three regions without further prevention and control measures. Besides, sensitivity analysis is carried out to illustrate the role of model parameters for TB control. Our finding reveals that appropriately improving the rate of timely treatment for actively infected people and increasing the rate of individuals with LTBI seeking preventive treatment could achieve the goal of TB elimination. In addition, another interesting finding shows that media impact can only reduce the number of active infections to a limited extent, but cannot change the prevalence of TB.

Dynamics and optimal control of tuberculosis model with the combined effects of vaccination, treatment and contaminated environments

Tuberculosis has affected human beings for thousands of years, and until today, tuberculosis still ranks third among 29 infectious diseases in China. However, most of the existing mathematical models consider a single factor, which is not conducive to the study of tuberculosis transmission dynamics. Therefore, this study considers the combined effects of vaccination, treatment, and contaminated environments on tuberculosis, and builds a new model with seven compartments of $ SVEITRW $ based on China's tuberculosis data. The study shows that when the basic reproduction number $ R_{0} $ is less than 1, the disease will eventually disappear, but when $ R_{0} $ is greater than 1, the disease may persist. In the numerical analysis part, we use Markov-chain Monte-Carlo method to obtain the optimal parameters of the model. Through the next generation matrix theory, we calculate that the $ R_{0} $ value of tuberculosis in China is $ 2.1102 $, that is, if not controlled, tuberculosis in China will not disappear over time. At the same time, through partial rank correlation coefficients, we find the most sensitive parameter to the basic reproduction number $ R_{0} $. On this basis, we combine the actual prevalence of tuberculosis in China, apply Pontryagin's maximum principle, and perform cost-effectiveness analysis to obtain the conditions required for optimal control. The analysis shows that four control strategies could effectively reduce the prevalence of TB, and simultaneously controlling $ u_{2}, u_{3}, u_{4} $ is the most cost-effective control strategy.

Using molecular testing and whole-genome sequencing for tuberculosis diagnosis in a low-burden setting: a cost-effectiveness analysis using transmission-dynamic modelling

BACKGROUND

Despite progress in TB control in low-burden countries like England and Wales, there are still diagnostic delays. Molecular testing and/or whole-genome sequencing (WGS) provide more rapid diagnosis but their cost-effectiveness is relatively unexplored in low-burden settings.

METHODS

An integrated transmission-dynamic health economic model is used to assess the cost-effectiveness of using WGS to replace culture-based drug-sensitivity testing, versus using molecular testing versus combined use of WGS and molecular testing, for routine TB diagnosis. The model accounts for the effects of faster appropriate treatment in reducing transmission, benefiting health and reducing future treatment costs. Cost-effectiveness is assessed using incremental net benefit (INB) over a 10-year horizon with a quality-adjusted life-year valued at £20 000, and discounting at 3.5% per year.

RESULTS

WGS shortens the time to drug sensitivity testing and treatment modification where necessary, reducing treatment and hospitalisation costs, with an INB of £7.1 million. Molecular testing shortens the time to TB diagnosis and treatment. Initially, this causes an increase in annual costs of treatment, but averting transmissions and future active TB disease subsequently, resulting in cost savings and health benefits to achieve an INB of £8.6 million (GeneXpert MTB/RIF) or £11.1 million (Xpert-Ultra). Combined use of Xpert-Ultra and WGS is the optimal strategy we consider, with an INB of £16.5 million.

CONCLUSION

Routine use of WGS or molecular testing is cost-effective in a low-burden setting, and combined use is the most cost-effective option. Adoption of these technologies can help low-burden countries meet the WHO End TB Strategy milestones, particularly the UK, which still has relatively high TB rates.

Data Analysis and Forecasting of Tuberculosis Prevalence Rates for Smart Healthcare Based on a Novel Combination Model

In recent years, healthcare has attracted much attention, which is looking for more and more data analytics in healthcare to relieve medical problems in medical staff shortage, ageing population, people living alone, and quality of life. Data mining, analysis, and forecasting play a vital role in modern social and medical fields. However, how to select a proper model to mine and analyze the relevant medical information in the data is not only an extremely challenging problem, but also a concerning problem. Tuberculosis remains a major global health problem despite recent and continued progress in prevention and treatment. There is no doubt that the effective analysis and accurate forecasting of global tuberculosis prevalence rates lay a solid foundation for the construction of an epidemic disease warning and monitoring system from a global perspective. In this paper, the tuberculosis prevalence rate time series for four World Bank income groups are targeted. Kruskal–Wallis analysis of variance and multiple comparison tests are conducted to determine whether the differences of tuberculosis prevalence rates for different income groups are statistically significant or not, and a novel combined forecasting model with its weights optimized by a recently developed artificial intelligence algorithm—cuckoo search—is proposed to forecast the hierarchical tuberculosis prevalence rates from 2013 to 2016. Numerical results show that the developed combination model is not only simple, but is also able to satisfactorily approximate the actual tuberculosis prevalence rate, and can be an effective tool in mining and analyzing big data in the medical field.

Progression from latent infection to active disease in dynamic tuberculosis transmission models: a systematic review of the validity of modelling assumptions

Mathematical modelling is commonly used to evaluate infectious disease control policy and is influential in shaping policy and budgets. Mathematical models necessarily make assumptions about disease natural history and, if these assumptions are not valid, the results of these studies can be biased. We did a systematic review of published tuberculosis transmission models to assess the validity of assumptions about progression to active disease after initial infection (PROSPERO ID CRD42016030009). We searched PubMed, Web of Science, Embase, Biosis, and Cochrane Library, and included studies from the earliest available date (Jan 1, 1962) to Aug 31, 2017. We identified 312 studies that met inclusion criteria. Predicted tuberculosis incidence varied widely across studies for each risk factor investigated. For population groups with no individual risk factors, annual incidence varied by several orders of magnitude, and 20-year cumulative incidence ranged from close to 0% to 100%. A substantial proportion of modelled results were inconsistent with empirical evidence: for 10-year cumulative incidence, 40% of modelled results were more than double or less than half the empirical estimates. These results demonstrate substantial disagreement between modelling studies on a central feature of tuberculosis natural history. Greater attention to reproducing known features of epidemiology would strengthen future tuberculosis modelling studies, and readers of modelling studies are recommended to assess how well those studies demonstrate their validity.

Disparities in model-based cost-effectiveness analyses of tuberculosis diagnosis: A systematic review

Background

Structural approach disparities were minimally addressed in past systematic reviews of model-based cost-effectiveness analyses addressing Tuberculosis management strategies. This review aimed to identify the structural approach disparities in model-based cost-effectiveness analysis studies addressing Tuberculosis diagnosis and describe potential hazards caused by those disparities.

Methods

A systematic search to identify studies published before October 2015 was performed in five electronic databases. After removal of duplication, studies’ titles and abstracts were screened based on predetermined criteria. The full texts of potentially relevant studies were subsequently screened and excluded when they did not address active pulmonary Tuberculosis diagnosis. Quality of the studies was assessed using the “Philips’ checklist.” Various data regarding general information, cost-effectiveness results, and disease modeling were extracted using standardized data extraction forms. Data pertaining to models’ structural approaches were compared and analyzed qualitatively for their applicability in various study settings, as well as their potential influence on main outcomes and cost-effectiveness conclusion.

Results

A total of 27 studies were included in the review. Most studies utilized a static model, which could underestimate the cost-effectiveness of the diagnostic tools strategies, due to the omission of indirect diagnosis effects, i.e. transmission reduction. A few structural assumption disparities were found in the dynamic models. Extensive disparities were found in the static models, consisting of varying structural assumptions regarding treatment outcomes, clinical diagnosis and empirical treatment, inpatient discharge decision, and re-diagnosis of false negative patients.

Conclusion

In cost-effectiveness analysis studies addressing active pulmonary Tuberculosis diagnosis, models showed numerous disparities in their structural approaches. Several structural approaches could be inapplicable in certain settings. Furthermore, they could contribute to under- or overestimation of the cost-effectiveness of the diagnosis tools or strategies. They could thus lead to ambiguities and difficulties when interpreting a study result. A set of recommendations is proposed to manage issues related to these structural disparities.

Comparison of two cash transfer strategies to prevent catastrophic costs for poor tuberculosis-affected households in low- and middle-income countries: An economic modelling study

BACKGROUND

Illness-related costs for patients with tuberculosis (TB) ≥20% of pre-illness annual household income predict adverse treatment outcomes and have been termed "catastrophic." Social protection initiatives, including cash transfers, are endorsed to help prevent catastrophic costs. With this aim, cash transfers may either be provided to defray TB-related costs of households with a confirmed TB diagnosis (termed a "TB-specific" approach); or to increase income of households with high TB risk to strengthen their economic resilience (termed a "TB-sensitive" approach). The impact of cash transfers provided with each of these approaches might vary. We undertook an economic modelling study from the patient perspective to compare the potential of these 2 cash transfer approaches to prevent catastrophic costs.

METHODS AND FINDINGS

Model inputs for 7 low- and middle-income countries (Brazil, Colombia, Ecuador, Ghana, Mexico, Tanzania, and Yemen) were retrieved by literature review and included countries' mean patient TB-related costs, mean household income, mean cash transfers, and estimated TB-specific and TB-sensitive target populations. Analyses were completed for drug-susceptible (DS) TB-related costs in all 7 out of 7 countries, and additionally for drug-resistant (DR) TB-related costs in 1 of the 7 countries with available data. All cost data were reported in 2013 international dollars ($). The target population for TB-specific cash transfers was poor households with a confirmed TB diagnosis, and for TB-sensitive cash transfers was poor households already targeted by countries' established poverty-reduction cash transfer programme. Cash transfers offered in countries, unrelated to TB, ranged from $217 to $1,091/year/household. Before cash transfers, DS TB-related costs were catastrophic in 6 out of 7 countries. If cash transfers were provided with a TB-specific approach, alone they would be insufficient to prevent DS TB catastrophic costs in 4 out of 6 countries, and when increased enough to prevent DS TB catastrophic costs would require a budget between $3.8 million (95% CI: $3.8 million-$3.8 million) and $75 million (95% CI: $50 million-$100 million) per country. If instead cash transfers were provided with a TB-sensitive approach, alone they would be insufficient to prevent DS TB-related catastrophic costs in any of the 6 countries, and when increased enough to prevent DS TB catastrophic costs would require a budget between $298 million (95% CI: $219 million-$378 million) and $165,367 million (95% CI: $134,085 million-$196,425 million) per country. DR TB-related costs were catastrophic before and after TB-specific or TB-sensitive cash transfers in 1 out of 1 countries. Sensitivity analyses showed our findings to be robust to imputation of missing TB-related cost components, and use of 10% or 30% instead of 20% as the threshold for measuring catastrophic costs. Key limitations were using national average data and not considering other health and social benefits of cash transfers.

CONCLUSIONS

A TB-sensitive cash transfer approach to increase all poor households' income may have broad benefits by reducing poverty, but is unlikely to be as effective or affordable for preventing TB catastrophic costs as a TB-specific cash transfer approach to defray TB-related costs only in poor households with a confirmed TB diagnosis. Preventing DR TB-related catastrophic costs will require considerable additional investment whether a TB-sensitive or a TB-specific cash transfer approach is used.

Analysis of Transmission and Control of Tuberculosis in Mainland China, 2005-2016, Based on the Age-Structure Mathematical Model

Tuberculosis (TB), an air-borne infectious disease, is a major public-health problem in China. The reported number of the active tuberculosis cases is about one million each year. The morbidity data for 2005–2012 reflect that the difference in morbidity based on age group is significant, thus the role of age-structure on the transmission of TB needs to be further developed. In this work, based on the reported data and the observed morbidity characteristics, we propose a susceptible-exposed-infectious-recovered (SEIR) epidemic model with age groupings, involving three categories: children, the middle-aged, and senior to investigate the role of age on the transmission of tuberculosis in Mainland China from 2005 to 2016. Then, we evaluated the parameters by the Least Square method and simulated the model and it had good alignment with the reported infected TB data in Mainland China. Furthermore, we estimated the basic reproduction number R0 of 1.7858, with an obtained 95% confidence interval for R0 of (1.7752,1.7963) by Latin hypercube sampling, and we completed a sensitivity analysis of R0 in terms of some parameters. Our study demonstrates that diverse age groups have different effects on TB. Two effective measures were found that would help reach the goals of the World Health Organization (WHO) End TB Strategy: an increase in the recovery rate and the reduction in the infectious rate of the senior age group.

Potential impact of spatially targeted adult tuberculosis vaccine in Gujarat, India

Some of the most promising vaccines in the pipeline for tuberculosis (TB) target adolescents and adults. Unlike for childhood vaccines, high-coverage population-wide vaccination is significantly more challenging for adult vaccines. Here, we aimed to estimate the impact of vaccine delivery strategies that were targeted to high-incidence geographical 'hotspots' compared with randomly allocated vaccination. We developed a spatially explicit mathematical model of TB transmission that distinguished these hotspots from the general population. We evaluated the impact of targeted and untargeted vaccine delivery strategies in India--a country that bears more than 25% of global TB burden, and may be a potential early adopter of the vaccine. We collected TB notification data and conducted a demonstration study in the state of Gujarat to validate our estimates of heterogeneity in TB incidence. We then projected the impact of randomly vaccinating 8% of adults in a single mass campaign to a spatially targeted vaccination preferentially delivered to 80% of adults in the hotspots, with both strategies augmented by continuous adolescent vaccination. In consultation with vaccine developers, we considered a vaccine efficacy of 60%, and evaluated the population-level impact after 10 years of vaccination. Spatial heterogeneity in TB notification (per 100,000/year) was modest in Gujarat: 190 in the hotspots versus 125 in the remaining population. At this level of heterogeneity, the spatially targeted vaccination was projected to reduce TB incidence by 28% after 10 years, compared with a 24% reduction projected to achieve via untargeted vaccination--a 1.17-fold augmentation in the impact of vaccination by spatially targeting. The degree of the augmentation was robust to reasonable variation in natural history assumptions, but depended strongly on the extent of spatial heterogeneity and mixing between the hotspot and general population. Identifying high-incidence hotspots and quantifying spatial mixing patterns are critical to accurate estimation of the value of targeted intervention strategies.

Improving Control of Tuberculosis in Low-Burden Countries: Insights from Mathematical Modeling

Tuberculosis control and elimination remains a challenge for public health even in low-burden countries. New technology and novel approaches to case-finding, diagnosis, and treatment are causes for optimism but they need to be used cost-effectively. This in turn requires improved understanding of the epidemiology of TB and analysis of the effectiveness and cost-effectiveness of different interventions. We describe the contribution that mathematical modeling can make to understanding epidemiology and control of TB in different groups, guiding improved approaches to public health interventions. We emphasize that modeling is not a substitute for collecting data but rather is complementary to empirical research, helping determine what are the key questions to address to maximize the public-health impact of research, helping to plan studies, and making maximal use of available data, particularly from surveillance, and observational studies. We provide examples of how modeling and related empirical research inform policy and discuss how a combination of these approaches can be used to address current questions of key importance, including use of whole-genome sequencing, screening and treatment for latent infection, and combating drug resistance.

Modelling tuberculosis trends in the USA

We present a mathematical transmission model of tuberculosis in the USA. The model is calibrated to recent trends of declining incidence in the US-born and foreign-born populations and is used in assessing relative impacts of treatment of latently infected individuals on elimination time, where elimination is defined as annual incidence <1 case/million. Provided current control efforts are maintained, elimination in the US-born population can be achieved before the end of this century. However, elimination in the foreign-born population is unlikely in this timeframe even with higher rates of targeted testing and treatment of residents of and immigrants to the USA with latent tuberculosis infection. Cutting transmission of disease as an interim step would shorten the time to elimination in the US-born population but foreign-born rates would remain above the elimination target.