Dynamics of a Mathematical Model for Tuberculosis with Variability in Susceptibility and Disease Progressions Due to Difference in Awareness Level

A mathematical fractal-fractional model to control tuberculosis prevalence with sensitivity, stability, and simulation under feasible circumstances

BACKGROUND

Tuberculosis, a global health concern, was anticipated to grow to 10.6 million new cases by 2021, with an increase in multidrug-resistant tuberculosis. Despite 1.6 million deaths in 2021, present treatments save millions of lives, and tuberculosis may overtake COVID-19 as the greatest cause of mortality. This study provides a six-compartmental deterministic model that employs a fractal-fractional operator with a power law kernel to investigate the impact of vaccination on tuberculosis dynamics in a population.

METHODS

Some important characteristics, such as vaccination and infection rate, are considered. We first show that the suggested model has positive bounded solutions and a positive invariant area. We evaluate the equation for the most important threshold parameter, the basic reproduction number, and investigate the model's equilibria. We perform sensitivity analysis to determine the elements that influence tuberculosis dynamics. Fixed-point concepts show the presence and uniqueness of a solution to the suggested model. We use the two-step Newton polynomial technique to investigate the effect of the fractional operator on the generalized form of the power law kernel.

RESULTS

The stability analysis of the fractal-fractional model has been confirmed for both Ulam-Hyers and generalized Ulam-Hyers types. Numerical simulations show the effects of different fractional order values on tuberculosis infection dynamics in society. According to numerical simulations, limiting contact with infected patients and enhancing vaccine efficacy can help reduce the tuberculosis burden. The fractal-fractional operator produces better results than the ordinary integer order in the sense of memory effect at diffract fractal and fractional order values.

CONCLUSION

According to our findings, fractional modeling offers important insights into the dynamic behavior of tuberculosis disease, facilitating a more thorough comprehension of their epidemiology and possible means of control.

A mathematical model for the co-dynamics of COVID-19 and tuberculosis

In this study, we formulated and analyzed a deterministic mathematical model for the co-infection of COVID-19 and tuberculosis, to study the co-dynamics and impact of each disease in a given population. Using each disease's corresponding reproduction number, the existence and stability of the disease-free equilibrium were established. When the respective threshold quantities R C , and R T are below unity, the COVID-19 and TB-free equilibrium are said to be locally asymptotically stable. The impact of vaccine (i.e., efficacy and vaccinated proportion) and the condition required for COVID-19 eradication was examined. Furthermore, the presence of the endemic equilibria of the sub-models is analyzed and the criteria for the phenomenon of backward bifurcation of the COVID-19 sub-model are presented. To better understand how each disease condition impacts the dynamics behavior of the other, we investigate the invasion criterion of each disease by computing the threshold quantity known as the invasion reproduction number. We perform a numerical simulation to investigate the impact of threshold quantities ( R C , R T ) with respect to their invasion reproduction number, co-infection transmission rate ( β c t ) , and each disease transmission rate ( β c , β t ) on disease dynamics. The outcomes established the necessity for the coexistence or elimination of both diseases from the communities. Overall, our findings imply that while COVID-19 incidence decreases with co-infection prevalence, the burden of tuberculosis on the human population increases.

A Mathematical Model of the Tuberculosis Epidemic

Tuberculosis has continued to retain its title as "the captain among these men of death". This is evident as it is the leading cause of death globally from a single infectious agent. TB as it is fondly called has become a major threat to the achievement of the sustainable development goals (SDG) and hence require inputs from different research disciplines. This work presents a mathematical model of tuberculosis. A compartmental model of seven classes was used in the model formulation comprising of the susceptible S, vaccinated V, exposed E, undiagnosed infectious I₁, diagnosed infectious I₂, treated T and recovered R. The stability analysis of the model was established as well as the condition for the model to undergo backward bifurcation. With the existence of backward bifurcation, keeping the basic reproduction number less than unity [Formula: see text] is no more sufficient to keep TB out of the community. Hence, it is shown by the analysis that vaccination program, diagnosis and treatment helps to control the TB dynamics. In furtherance to that, it is shown that preference should be given to diagnosis over treatment as diagnosis precedes treatment. It is as well shown that at lower vaccination rate (0-20%), TB would still be endemic in the population. As such, high vaccination rate is required to send TB out of the community.

Scenario analysis for programmatic tuberculosis control in Bangladesh: a mathematical modelling study

Tuberculosis (TB) is a major public health problem in Bangladesh. Although the National TB control program of Bangladesh is implementing a comprehensive expansion of TB control strategies, logistical challenges exist, and there is significant uncertainty concerning the disease burden. Mathematical modelling of TB is considered one of the most effective ways to understand the dynamics of infection transmission and allows quantification of parameters in different settings, including Bangladesh. In this study, we present a two-strain mathematical modelling framework to explore the dynamics of drug-susceptible (DS) and multidrug-resistant (MDR) TB in Bangladesh. We calibrated the model using DS and MDR-TB annual incidence data from Bangladesh from years 2001 to 2015. Further, we performed a sensitivity analysis of the model parameters and found that the contact rate of both strains had the largest influence on the basic reproduction numbers [Formula: see text] and [Formula: see text] of DS and MDR-TB, respectively. Increasingly powerful intervention strategies were developed, with realistic impact and coverage determined with the help of local staff. We simulated for the period from 2020 to 2035. Here, we projected the DS and MDR-TB burden (as measured by the number of incident cases and mortality) under a range of intervention scenarios to determine which of these scenario is the most effective at reducing burden. Of the single-intervention strategies, enhanced case detection is the most effective and prompt in reducing DS and MDR-TB incidence and mortality in Bangladesh and that with GeneXpert testing was also highly effective in decreasing the burden of MDR-TB. Our findings also suggest combining additional interventions simultaneously leads to greater effectiveness, particularly for MDR-TB, which we estimate requires a modest investment to substantially reduce, whereas DS-TB requires a strong sustained investment.

Modeling drug-resistant tuberculosis amplification rates and intervention strategies in Bangladesh

Tuberculosis (TB) is the seventh leading cause of morbidity and mortality in Bangladesh. Although the National TB control program (NTP) of Bangladesh is implementing its nationwide TB control strategies, more specific and effective single or combination interventions are needed to control drug-susceptible (DS) and multi-drug resistant (MDR) TB. In this study, we developed a two strain TB mathematical model with amplification and fit it to the Bangladesh TB data to understand the transmission dynamics of DS and MDR TB. Sensitivity analysis was used to identify important parameters. We evaluated the cost-effectiveness of varying combinations of four basic control strategies including distancing, latent case finding, case holding and active case finding, all within the optimal control framework. From our fitting, the model with different transmission rates between DS and MDR TB best captured the Bangladesh TB reported case counts. The estimated basic reproduction number for DS TB was 1.14 and for MDR TB was 0.54, with an amplification rate of 0.011 per year. The sensitivity analysis also indicated that the transmission rates for both DS and MDR TB had the largest influence on prevalence. To reduce the burden of TB (both DS and MDR), our finding suggested that a quadruple control strategy that combines distancing control, latent case finding, case holding and active case finding is the most cost-effective. Alternative strategies can be adopted to curb TB depending on availability of resources and policy makers’ decisions.

Non-vaccine strategies for cholera prevention and control: India's preparedness for the global roadmap

BACKGROUND

Recently World Health Organization's Global Task Force on Cholera Control (GTFCC) has published a global roadmap for prevention and control of cholera. We review preparedness of existing governmental non-vaccine programs and strategies for cholera prevention and control in India. We also describe strengths and gaps in the context of implementation of the global roadmap.

METHODS

We reviewed published literature on non-vaccine based strategies for prevention and control of cholera in India and analyzed strengths and weaknesses of Government of India's major anti-cholera and ante-diarrhea initiatives under Integrated Disease Surveillance Program (IDSP), National Rural Health Mission (NRHM), and other disease surveillance platforms.

RESULTS

The first strategy of the WHO global roadmap, namely, preparedness for early detection and outbreak containment, has been addressed by the IDSP. NRHM complements IDSP activities by focusing on sanitation, hygiene, nutrition, and safe drinking water. We identified the need to adopt stricter case definitions and data validation protocols. Multi-sectoral approach to prevent cholera occurrences and re-occurrences [the second suggested strategy in the global roadmap], highlights identification of hotspots and implementing strategies based on transmission dynamics. We recommend development of comprehensive models by integrating data sources beyond the national programs to eliminate cholera hotspots in India. Implementing the third proposed strategy in the global roadmap, coordinated technical support, resource mobilization, and partnerships at local and global levels, has major challenges in India due to structural issues related to health systems and health programs.

CONCLUSION

Even with a robust public health infrastructure, absence of a national cholera program might have resulted in lack of specific focus and concerted efforts for cholera prevention and control in India. A National Taskforce for Cholera Control must develop India-specific 'National Cholera Prevention and Response Road Map' with an appropriate administrative and financially viable framework for its implementation.

Mathematical analysis of a tuberculosis model with imperfect vaccine

Since 1921, the Bacille Calmette–Guerin (BCG) vaccine continues to be the most widely used vaccine for the prevention of Tuberculosis (TB). However, the immunity induced by BCG wanes out after some time making the vaccinated individual susceptible to TB infection. In this work, we formulate a mathematical model that incorporates the vaccination of newly born children and older susceptible individuals in the transmission dynamics of TB in a population, with a vaccine that can confer protection on older susceptible individuals. In the absence of disease-induced deaths, the model is shown to undergo the phenomenon of backward bifurcation where a stable disease-free equilibrium (DFE) co-exists with a stable positive (endemic) equilibrium when the associated reproduction number is less than unity. It is shown that this phenomenon does not exist in the absence of imperfect vaccine, exogenous reinfection, and reinfection of previously treated individuals. It is further shown that a special case of the model has a unique endemic equilibrium point (EEP), which is globally asymptotically stable when the associated reproduction number exceeds unity. Uncertainty and sensitivity analysis are carried out to identify key parameters that have the greatest influence on the transmission dynamics of TB in the population using the total population of latently infected individuals, total number of actively infected individuals, disease incidence, and the effective reproduction number as output responses. The analysis shows that the top five parameters of the model that have the greatest influence on the effective reproduction number of the model are the transmission rate, the fraction of fast disease progression, modification parameter which accounts for reduced likelihood to infection by vaccinated individuals due to imperfect vaccine, rate of progression from latent to active TB, and the treatment rate of actively infected individuals, with other key parameters influencing the outcomes of the other output responses. Numerical simulations suggest that with higher vaccination rate of older susceptible individuals, fewer new born children need to be vaccinated, in order to achieve disease eradication.

Heterogeneous infectiousness in mathematical models of tuberculosis: A systematic review

TB mathematical models employ various assumptions and approaches in dealing with the heterogeneous infectiousness of persons with active TB. We reviewed existing approaches and considered the relationship between them and existing epidemiological evidence. We searched the following electronic bibliographic databases from inception to 9 October 2018: MEDLINE, EMBASE, Biosis, Global Health and Scopus. Two investigators extracted data using a standardised data extraction tool. We included in the review any transmission dynamic model of M. tuberculosis transmission explicitly simulating heterogeneous infectiousness of person with active TB. We extracted information including: study objective, model structure, number of active TB compartments, factors used to stratify the active TB compartment, relative infectiousness of each active TB compartment and any intervention evaluated in the model. Our search returned 1899 unique references, of which the full text of 454 records were assessed for eligibility, and 99 studies met the inclusion criteria. Of these, 89 used compartmental models implemented with ordinary differential equations, while the most common approach to stratification of the active TB compartment was to incorporate two levels of infectiousness. However, various clinical characteristics were used to stratify the active TB compartments, and models differed as to whether they permitted transition between these states. Thirty-four models stratified the infectious compartment according to sputum smear status or pulmonary involvement, while 18 models stratified based on health care-related factors. Variation in infectiousness associated with drug-resistant M. tuberculosis was the rationale for stratifying active TB in 33 models, with these models consistently assuming that drug-resistant active TB cases were less infectious. Given the evidence of extensive heterogeneity in infectiousness of individuals with active TB, an argument exists for incorporating heterogeneous infectiousness, although this should be considered in light of the objectives of the study and the research question. PROSPERO Registration: CRD42019111936.

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

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.

Mathematical Analysis of the Transmission Dynamics of HIV Syphilis Co-infection in the Presence of Treatment for Syphilis

The re-emergence of syphilis has become a global public health issue, and more persons are getting infected, especially in developing countries. This has also led to an increase in the incidence of human immunodeficiency virus (HIV) infections as some studies have shown in the recent decade. This paper investigates the synergistic interaction between HIV and syphilis using a mathematical model that assesses the impact of syphilis treatment on the dynamics of syphilis and HIV co-infection in a human population where HIV treatment is not readily available or accessible to HIV-infected individuals. In the absence of HIV, the syphilis-only model undergoes the phenomenon of backward bifurcation when the associated reproduction number ([Formula: see text]) is less than unity, due to susceptibility to syphilis reinfection after recovery from a previous infection. The complete syphilis-HIV co-infection model also undergoes the phenomenon of backward bifurcation when the associated effective reproduction number ([Formula: see text]) is less than unity for the same reason as the syphilis-only model. When susceptibility to syphilis reinfection after treatment is insignificant, the disease-free equilibrium of the syphilis-only model is shown to be globally asymptotically stable whenever the associated reproduction number ([Formula: see text]) is less than unity. Sensitivity and uncertainty analysis show that the top three parameters that drive the syphilis infection (with respect to the associated response function, [Formula: see text]) are the contact rate ([Formula: see text]), modification parameter that accounts for the increased infectiousness of syphilis-infected individuals in the secondary stage of the infection ([Formula: see text]) and treatment rate for syphilis-only infected individuals in the primary stage of the infection ([Formula: see text]). The co-infection model was numerically simulated to investigate the impact of various treatment strategies for primary and secondary syphilis, in both singly and dually infected individuals, on the dynamics of the co-infection of syphilis and HIV. It is observed that if concerted effort is exerted in the treatment of primary and secondary syphilis (in both singly and dually infected individuals), especially with high treatment rates for primary syphilis, this will result in a reduction in the incidence of HIV (and its co-infection with syphilis) in the population.