To treat or not to treat: the case of tuberculosis

A framework for the modelling and the analysis of epidemiological spread in commuting populations

In the present paper, our goal is to establish a framework for the mathematical modelling and the analysis of the spread of an epidemic in a large population commuting regularly, typically along a time-periodic pattern, as is roughly speaking the case in populous urban centre. Our modelling contribution develops along two axes. To model the commuting, we consider a large number of distinct homogeneous groups of individuals of various sizes, called subpopulations, and focus on the modelling of the changing conditions of their mixing along time and of the induced disease transmission. Also, for the purposes of the study, we propose a general class of epidemiological models in which the 'force of infection' plays a central role, which extends and unifies several classes previously developed. We take special care in explaining the modelling approach in details, and provide first analytic results that allow to compute or estimate the value of the basic reproduction number for such general periodic epidemic systems.

Mathematical modeling of cholera dynamics in the presence of antimicrobial utilization strategy

Antimicrobial resistance poses a significant threat to public health, particularly in cholera treatment. The emergence of antibiotic resistance, coupled with the sharp decline in pharmaceutical companies developing new cholera antibiotics, is a cause for concern. We formulate a multidrug-resistant (MDR) cholera epidemic model that incorporates a stage-switching strategy between two antibiotics to reduce the magnitude of resistance. The model is analyzed mathematically, and sensitivity analysis of the reproduction number is performed using sub-reproduction numbers. Stability analysis of the cholera-sensitive-only and cholera-resistant-only equilibria is investigated using Centre Manifold Theory. The model is calibrated through Markov Chain Monte Carlo simulations in Stan, showing stability at equilibrium points, which is further verified through numerical simulations. The simulations demonstrate an inverse relationship between the number of MDR cholera cases and the number of individuals receiving second-line treatment for cholera. This study suggests that the correct use of antibiotics can effectively manage the emergence of antimicrobial resistance. From a public health policy perspective, these findings emphasize the importance of antibiotic stewardship programs and the need for policies that promote the responsible use of existing antibiotics while encouraging the development of new treatment options. Such measures could help mitigate the global burden of MDR cholera and prevent further escalation of resistance.

Global dynamics of discrete mathematical models of tuberculosis

In this paper, we develop discrete models of Tuberculosis (TB). This includes SEI endogenous and exogenous models without treatment. These models are then extended to a SEIT model with treatment. We develop two types of net reproduction numbers, one is the traditional R 0 which is based on the disease-free equilibrium, and a new net reproduction number R 0 ( E ∗ ) based on the endemic equilibrium. It is shown that the disease-free equilibrium is globally asymptotically stable if R 0 ≤   1 and unstable if R 0 > 1 . Moreover, the endemic equilibrium is locally asymptotically stable if R 0 ( E ∗ ) < 1 < R 0 .

Analysis of a fractional order epidemiological model for tuberculosis transmission with vaccination and reinfection

This study has been carried out using a novel mathematical model on the dynamics of tuberculosis (TB) transmission considering vaccination, endogenous re-activation of the dormant infection, and exogenous re-infection. We can comprehend the behavior of TB under the influence of vaccination from this article. We compute the basic reproduction number ( R 0 ) as well as the vaccination reproduction number ( R v ) using the next-generation matrix (NGM) approach. The theoretical analysis demonstrates that the disease-free equilibrium point is locally asymptotically stable, and the fractional order system is Ulam-Hyers type stable. We perform numerical simulation of our model using the Adams-Bashforth 3-step method to verify the theoretical results and to show the model outputs graphically. By performing data fitting, we observe that our formulated model produces results that closely match real-world data. Our findings indicate that vaccinating a limited segment of the population can effectively eradicate the disease. The numerical simulations also show that vaccination can reduce the number of susceptible and infectious individuals in the population. Moreover, the graphical representations illustrate that the number of infected individuals rises due to both exogenous reinfection and endogenous reactivation.

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.

A two-strain model of infectious disease spread with asymmetric temporary immunity periods and partial cross-immunity

We introduce a two-strain model with asymmetric temporary immunity periods and partial cross-immunity. We derive explicit conditions for competitive exclusion and coexistence of the strains depending on the strain-specific basic reproduction numbers, temporary immunity periods, and degree of cross-immunity. The results of our bifurcation analysis suggest that, even when two strains share similar basic reproduction numbers and other epidemiological parameters, a disparity in temporary immunity periods and partial or complete cross-immunity can provide a significant competitive advantage. To analyze the dynamics, we introduce a quasi-steady state reduced model which assumes the original strain remains at its endemic steady state. We completely analyze the resulting reduced planar hybrid switching system using linear stability analysis, planar phase-plane analysis, and the Bendixson-Dulac criterion. We validate both the full and reduced models with COVID-19 incidence data, focusing on the Delta (B.1.617.2), Omicron (B.1.1.529), and Kraken (XBB.1.5) variants. These numerical studies suggest that, while early novel strains of COVID-19 had a tendency toward dramatic takeovers and extinction of ancestral strains, more recent strains have the capacity for co-existence.

Nonlinear neural networks adaptive control for a class of fractional-order tuberculosis model

The problem of nonlinear adaptive control for a class of fractional-order tuberculosis (TB) model is studied in this paper. By analyzing the transmission mechanism of TB and the characteristics of fractional calculus, a fractional-order TB dynamical model is established with media coverage and treatment as control variables. With the help of universal approximation principle of radial basis function neural networks and the positive invariant set of established TB model, the expressions of control variables are designed and the stability of error model is analyzed. Thus, the adaptive control method can guarantee that the number of susceptible and infected individuals can be kept close to the corresponding control targets. Finally, the designed control variables are illustrated by numerical examples. The results indicate that the proposed adaptive controllers can effectively control the established TB model and ensure the stability of controlled model, and two control measures can protect more people from tuberculosis infection.

Global Stability for an Endogenous-Reactivated Tuberculosis Model with Beddington-DeAngelis Incidence, Distributed Delay and Relapse

A tuberculosis (TB) epidemic model with Beddington-DeAngelis incidence and distributed delay is proposed to characterize the interaction between latent period, endogenous reactivation, treatment of latent TB infection, as well as relapse. The basic reproduction number R 0 is defined, and the globally asymptotic stability of disease-free equilibrium is shown when R 0 < 1 , while if   R 0 > 1   the globally asymptotic stability of endemic equilibrium is also acquired. Theoretical results are validated through performing numerical simulations, wherein we detect that TB dynamic behavior between models with discrete and distributed delays could be same and opposite, and TB is more persistent in the model with distributed delay. Besides, increasing the protection level of susceptible and infectious individuals is crucial for the control of TB.

A hospital demand and capacity intervention approach for COVID-19

The mathematical interpretation of interventions for the mitigation of epidemics in the literature often involves finding the optimal time to initiate an intervention and/or the use of the number of infections to manage impact. Whilst these methods may work in theory, in order to implement effectively they may require information which is not likely to be available in the midst of an epidemic, or they may require impeccable data about infection levels in the community. In reality, testing and cases data can only be as good as the policy of implementation and the compliance of the individuals, which implies that accurately estimating the levels of infections becomes difficult or complicated from the data that is provided. In this paper, we demonstrate a different approach to the mathematical modelling of interventions, not based on optimality or cases, but based on demand and capacity of hospitals who have to deal with the epidemic on a day to day basis. In particular, we use data-driven modelling to calibrate a susceptible-exposed-infectious-recovered-died type model to infer parameters that depict the dynamics of the epidemic in several regions of the UK. We use the calibrated parameters for forecasting scenarios and understand, given a maximum capacity of hospital healthcare services, how the timing of interventions, severity of interventions, and conditions for the releasing of interventions affect the overall epidemic-picture. We provide an optimisation method to capture when, in terms of healthcare demand, an intervention should be put into place given a maximum capacity on the service. By using an equivalent agent-based approach, we demonstrate uncertainty quantification on the likelihood that capacity is not breached, by how much if it does, and the limit on demand that almost guarantees capacity is not breached.

A synthesized model of tuberculosis transmission featuring treatment abandonment

In this paper, we propose and justify a synthesized version of the tuberculosis transmission model featuring treatment abandonment. In contrast to other models that account for the treatment abandonment, our model has only four state variables or classes (susceptible, latent, infectious, and treated), while people abandoning treatment are not gathered into an additional class. The proposed model retains the core properties that are highly desirable in epidemiological modeling. Namely, the disease transmission dynamics is characterized by the basic reproduction number $ \mathscr{R}_0 $, a threshold value that determines the number of possible steady states and their stability properties. It is shown that the disease-free equilibrium is globally asymptotically stable (GAS) only if $ \mathscr{R}_0 < 1 $, while a strictly positive endemic equilibrium exists and is GAS only if $ \mathscr{R}_0 > 1. $ Analysis of the dependence of $ \mathscr{R}_0 $ on the treatment abandonment rate shows that a reduction of the treatment abandonment rate has a positive effect on the disease incidence and results in avoiding disease-related fatalities.

Epidemic Dynamics of Two-Pathogen Spreading for Pairwise Models

In the real world, pathogens do not exist in isolation. The transmission of one pathogen may be affected by the presence of other pathogens, and certain pathogens generate multiple strains with different spreading features. Hence, the behavior of multi-pathogen transmission has attracted much attention in epidemiological research. In this paper, we use the pairwise approximation method to formulate two-pathogen models capturing cross-immunity, super-infection, and co-infection phenomena, in which each pathogen follows a susceptible-infected-susceptible (SIS) mechanism. For each model, we calculate the basic reproduction number and analyze the stability of equilibria, and discuss the differences from the mean-field approach. We demonstrate that simulations are in good agreement with the analytical results.

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.

Bifurcation analysis of a model of tuberculosis epidemic with treatment of wider population suggesting a possible role in the seasonality of this disease

In this paper, a novel epidemiological model describing the evolution of tuberculosis in a human population is proposed. This model is of the form SEIR, where S stands for Susceptible people, E for Exposed (infected but not yet infectious) people, I for Infectious people, and R for Recovered people. The main characteristic of this model inspired from the disease biology and some realistic hypothesis is that the treatment is administered not only to infectious but also to exposed people. Moreover, this model is characterized by an open structure, as it considers the transfer of infected or infectious people to other regions more conducive to their care and accepts treatment for exposed or infectious patients coming from other regions without care facilities. Stability and bifurcation of the solutions of this model are investigated. It is found that saddle-focus bifurcation occurs when the contact parameter β passes through some critical values. The model undergoes a Hopf bifurcation when the quality of treatment r is considered as a bifurcation parameter. It is shown also that the system exhibits saddle-node bifurcation, which is a transcritical bifurcation between equilibrium points. Numerical simulations are done to illustrate these theoretical results. Amazingly, the Hopf bifurcation suggests an unexpected and never suggested explanation of seasonality of such a disease, linked to the quality of treatment.

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.

A mathematical analysis of a tuberculosis epidemic model with two treatments and exogenous re-infection

In this paper, we propose a mathematical model of tuberculosis with two treatments and exogenous re-infection, in which the treatment is effective for a number of infectious individuals and it fails for some other infectious individuals who are being treated. We show that the model exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibria when the related basic reproduction number is less than unity. Also, it is shown that under certain conditions the model cannot exhibit backward bifurcation. Furthermore, it is shown in the absence of re-infection, the backward bifurcation phenomenon does not exist, in which the disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity. The global asymptotic stability of the endemic equilibrium, when the associated reproduction number is greater than unity, is established using the geometric approach. Numerical simulations are presented to illustrate our main results.

A structured Markov chain model to investigate the effects of pre-exposure vaccines in tuberculosis control

In this paper, the interest is in a structured Markov chain model to describe the transmission dynamics of tuberculosis (TB) in the setting of small communities of hosts sharing confined spaces, and to explore the potential impact of new pre-exposure vaccines on reducing the number of new TB cases during an outbreak of the disease. The model under consideration incorporates endogenous reactivation of latent tubercle bacilli, exogenous reinfection of latently infected TB hosts, loss of effectiveness of the vaccine protection, and death of hosts due to tubercle bacilli and from causes beyond TB. Various probabilistic measures are defined and analytically studied to describe extreme values and the number of vaccinations during an outbreak, and a random version of the basic reproduction number is used to measure the transmission potential during the initial phase of the epidemic. Our numerical experiments allow us to compare different pre-exposure vaccines versus the level of coverage in terms of these probabilistic measures.

The prevention and control of tuberculosis: an analysis based on a tuberculosis dynamic model derived from the cases of Americans

Background

Tuberculosis (TB), a preventable and curable disease, is claimed as the second largest number of fatalities, and there are 9,025 cases reported in the United States in 2018. Many researchers have done a lot of research and achieved remarkable results, but TB is still a severe problem for human beings. The study is a further exploration of the prevention and control of tuberculosis.

Methods

In the paper, we propose a new dynamic model to study the transmission dynamics of TB, and then use global differential evolution and local sequential quadratic programming (DESQP) optimization algorithm to estimate parameters of the model. Finally, we use Latin hypercube sampling (LHS) and partial rank correlation coefficients (PRCC) to analyze the influence of parameters on the basic reproduction number (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal R_{0}$\end{document}R0) and the total infectious (including the diagnosed, undiagnosed and incomplete treatment infectious), respectively.

Results

According to the research, the basic reproduction number is computed as 2.3597 from 1984 to 2018, which means TB is also an epidemic in the US. The diagnosed rate is 0.6082, which means the undiagnosed will be diagnosed after 1.6442 years. The diagnosed will recover after an average of 1.9912 years. Moreover, some diagnosed will end the treatment after 1.7550 years for some reason. From the study, it’s shown that 2.40% of the recovered will be reactivated, and 13.88% of the newborn will be vaccinated. However, the immune system will be lost after about 19.6078 years.

Conclusion

Through the results of this study, we give some suggestions to help prevent and control the TB epidemic in the United States, such as prolonging the protection period of the vaccine by developing new and more effective vaccines to prevent TB; using the Chemoprophylaxis for incubation patients to prevent their conversion into active TB; raising people’s awareness of the prevention and control of TB and treatment after illness; isolating the infected to reduce the spread of TB. According to the latest report in the announcement that came at the first WHO Global Ministerial Conference on Ending tuberculosis in the Sustainable Development Era, we predict that it is challenging to control TB by 2030.

Country-specific intervention strategies for top three TB burden countries using mathematical model

Tuberculosis (TB) is one of the top 10 causes of death globally and the leading cause of death by a single infectious pathogen. The World Health Organization (WHO) has declared the End TB Strategy, which targets a 90% reduction in the incidence rate by the year 2035 compared to the level in the year 2015. In this work, a TB model is considered to understand the transmission dynamics in the top three TB burden countries-India, China, and Indonesia. Country-specific epidemiological parameters were identified using data reported by the WHO. If India and Indonesia succeed in enhancing their treatment protocols and increase treatment and treatment success rate to that of China, the incidence rate could be reduced by 65.99% and 68.49%, respectively, by the end of 2035. Evidently, complementary interventions are essential to achieve the WHO target. Our analytical approach utilizes optimal control theory to obtain time-dependent nonpharmaceutical and latent case finding controls. The objective functional of the optimal control problem includes a payoff term reflecting the goal set by WHO. Appropriate combinations of control strategies are investigated. Based on the results, gradual enhancement and continuous implementation of intervention measures are recommended in each country.

Optimal control analysis of vector-host model with saturated treatment

Vector-host infectious diseases remain a challenging issue and cause millions of deaths each year globally. In such outbreaks, many countries especially developing or underdevelopment faces a situation where the number of infected individuals is getting larger and the medical facilities are limited. In this paper, we construct an epidemic model to explore the transmission dynamics of vector-borne diseases with nonlinear saturated incidence rate and saturated treatment function. This type of incidence rate, as well as the saturated treatment function, is also known as the Holling type II form and describes the effect of delayed treatment. Initially, we formulate a mathematical model and then present the basic analysis of the model including the positivity and boundedness of the solution. The threshold quantity R 0 is presented and the stability analysis of the system is carried out for the model equilibria. The global stability results are shown using the Lyapunov function of Goh-Voltera type. The existence of backward bifurcation is discussed using the central manifold theory. Further, the global sensitivity analysis of the model is carried out using the Latin Hypercube sampling and the partial rank correlation coefficient techniques. Moreover, an optimal control problem is formulated and the necessary optimality conditions are investigated in order to eradicate the disease in a community. Four strategies are presented by choosing different set of controls combination for the disease minimization. Finally, the numerical simulations of each strategy are depicted to demonstrate the importance of suggesting control interventions on the disease dynamics and eradication.

GLOBAL DYNAMICS OF A MATHEMATICAL MODEL OF TUBERCULOSIS WITH AGE-DEPENDENT LATENCY AND ACTIVE INFECTION

In this paper, mathematical analysis is carried out for a mathematical model of Tuberculosis (TB) with age-dependent latency and active infection. The model divides latent TB infection into two stages: an early stage of high risk of developing active TB and a late stage of lower risk for developing active TB. Infected persons initially progress through the early latent TB stage and then can either progress to active TB infection or progress to late latent TB infection. The model is formulated by incorporating the duration that an individual has spent in the stages of the early latent TB, the late latent TB and the active TB infection as variables. By constructing suitable Lyapunov functionals and using LaSalle’s invariance principle, it is shown that the global dynamics of the disease is completely determined by the basic reproduction number: if the basic reproduction number is less than unity, the TB always dies out; if the basic reproduction number is greater than unity, a unique endemic steady state exists and is globally asymptotically stable in the interior of the feasible region and therefore the TB becomes endemic. Numerical simulations are carried out to illustrate the theoretical results.

Global dynamics of a tuberculosis model with fast and slow progression and age-dependent latency and infection

In this paper, a mathematical model describing tuberculosis transmission with fast and slow progression and age-dependent latency and infection is investigated. It is assumed in the model that infected individuals can develop tuberculosis by either of two pathogenic mechanisms: direct progression or endogenous reactivation. It is shown that the transmission dynamics of the disease is fully determined by the basic reproduction number. By analyzing corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state of the model is established. By using the persistence theory for infinite dimensional system, it is proved that the system is uniformly persistent when the basic reproduction number is greater than unity. By constructing suitable Lyapunov functionals and using LaSalle's invariance principle, it is verified that the global dynamics of the system is completely determined by the basic reproduction number.

Nonlinear adaptive control of tuberculosis with consideration of the risk of endogenous reactivation and exogenous reinfection

Tuberculosis is one of deadly diseases in many countries that attacks to the human body and causes damage to the lung, causing bloody coughing and if left untreated, it will kill half of the affected people. Tuberculosis bacteria can stay latent and reactivate after passing appropriate conditions. For this reason, control of this disease and treatment of infected people has a significant importance, and observing health issues can prevent the spread of it. In this paper, a nonlinear adaptive control method is proposed for the first time in order to control and treat tuberculosis outbreak subjected to the modeling uncertainty. To design a control system being robust against uncertainties, an adaptation law is defined to update values of estimated parameters and ensures the whole system stability. The treatment achievement and stability of the closed-loop system is proved by the Lyapunov theorem and confirmed by some simulations. The proposed strategy has the capability to control the tuberculosis outbreak by reducing the numbers of active infectious and persistent latent individuals based on their desired values in the society.

Effect of mixed infection on TB dynamics

Poor living conditions, overcrowding and strain diversity are some of the factors that influence mixed infection in Tuberculosis (TB) at the population level. We formulate a mathematical model for mixed infection in TB using nonlinear ordinary differential equations where such factors were represented as probabilities of acquiring mixed infection. A qualitative analysis of the model shows that it exhibits multiple endemic equilibria and backward bifurcation for certain parameter values. The reactivation rate and transmission rate of individuals with mixed infection were of importance as well as the probabilities for latent individuals to acquire mixed infection. We calculate the prevalence of mixed infection from the model and the effect of mixed infection on TB incidence, TB prevalence and Mycobacterium tuberculosis (MTB) infection rate. Numerical simulations show that mixed infection may explain high TB incidences in areas which have a high strain diversity, poor living conditions and are overcrowded even without HIV.

Endemic Disease Models

In this chapter, we consider models for disease that may be endemic. In the preceding chapter we studied SIS models with and without demographics and SIR models with demographics. In each model, the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0$$ \end{document}R0 determined a threshold. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0 < 1$$ \end{document}R0<1 the disease dies out, while if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0 > 1$$ \end{document}R0>1 the disease becomes endemic. The analysis in each case involves determination of equilibria and determining the asymptotic stability of each equilibrium by linearization about the equilibrium. In each of the cases studied in the preceding chapter the disease-free equilibrium was asymptotically stable if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0 < 1$$ \end{document}R0<1 and if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0 > 1$$ \end{document}R0>1 there was a unique endemic equilibrium that was asymptotically stable. In this chapter, we will see that these properties continue to hold for many more general models, but there are situations in which there may be an asymptotically stable endemic equilibrium when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0 < 1$$ \end{document}R0<1, and other situations in which there is an endemic equilibrium that is unstable for some values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal {R}_0 > 1$$ \end{document}R0>1.

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.

Dynamics between infectious diseases with two susceptibility conditions: A mathematical model

This paper presents a novel epidemiological transmission model of a population affected by two different susceptible-infected-susceptible infectious diseases. For each disease, individuals fall into one of the two susceptibility conditions in which one of the diseases has the highest occurrence level. This model is unique in assuming that: (a) if an individual is infected by one disease, their susceptibility to the other disease is increased; (b) when an individual recovers from a disease they become less susceptible to it, i.e. they acquire partial immunity. The model captures these two assumptions by utilizing a coupled system of differential equations. Dynamic analysis of the system is based on basic reproductive number theory, and pattern visualization was performed using numerical simulation.

Systems thinking and ethics in public health: a necessary and mutually beneficial partnership

Systems thinking has emerged as a means of conceptualizing and addressing complex public health problems, thereby challenging more commonplace understanding of problems and corresponding solutions as straightforward explanations of cause and effect. Systems thinking tries to address the complexity of problems through qualitative and quantitative modeling based on a variety of systems theories, each with their own assumptions and, more importantly, implicit and unexamined values. To date, however, there has been little engagement between systems scientists and those working in bioethics and public health ethics. The goal of this paper is to begin to consider what it might mean to combine systems thinking with public health ethics to solve public health challenges. We argue that there is a role for ethics in systems thinking in public health as a means of elucidating implicit assumptions and facilitating ethics debate and dialogue with key stakeholders.

Systematic evaluation of the population-level effects of alternative treatment strategies on the basic reproduction number

An approach to estimate the influence of the treatment-type controls on the basic reproduction number, R₀, is proposed and elaborated. The presented approach allows one to estimate the effect of a given treatment strategy or to compare a number of different treatment strategies on the basic reproduction number. All our results are valid for sufficiently small values of the control. However, in many cases it is possible to extend this analysis to larger values of the control as was illustrated by examples.

Dynamic model of tuberculosis considering multi-drug resistance and their applications

Infectious diseases have always been a problem that threatens people's health and tuberculosis is one of the major. With the development of medical scientific research, drug-resistant infectious diseases have become a more intractable threat because various drugs and antibiotics are widely used in the process of fighting against infectious diseases. In this paper, an improved dynamic model of infectious diseases considering population dynamics and drug resistance is established. The feasible region, equilibrium points and stability of the model are analyzed. Based on the existing data, this model can predict the development of the epidemic situation through numerical simulation, and put forward some relevant measures and suggestions.

Modelling heterogeneity in host susceptibility to tuberculosis and its effect on public health interventions

A tuberculosis (TB) model that accounts for heterogeneity in host susceptibility to tuberculosis is proposed, with the aim of investigating the implications this may have for the effectiveness of public health interventions. The model examines the possibility that recovered individuals treated from active TB and individuals treated with preventive therapy acquire different levels of immunity. This contrasts with recent studies that assume the two cohorts acquire the same level of immunity, and therefore both groups are reinfected at the same rate. The analysis presented here examines the impact of this assumption when designing intervention strategies. Comparison of reinfection rates between cohorts treated with preventive therapy and recovered individuals who were previously treated for active TB provides important epidemiological insights. It is found that the reinfection rate of the cohort treated with preventive therapy is the one that plays the key role in qualitative changes in TB dynamics. By contrast, the reinfection rate of recovered individuals (previously treated from active TB) plays a minor role. Moreover, the study shows that preventive treatment of individuals during early latency is always beneficial regardless of the level of susceptibility to reinfection. Further, if patients have greater immunity following treatment for late latent infection, then treatment is again beneficial. However, if susceptibility increases following treatment for late latent infection, the effect of treatment depends on the epidemiological setting. That is: (i) in (very) low burden settings, the effect on reactivation predominates and the burden declines with treatment; (ii) in moderate to high burden settings the effect of reinfection predominates and burden increases with treatment. The effect is most dominant between the two reinfection thresholds, RT2 and RT1, respectively associated with individuals being treated with preventive therapy and individuals with untreated late latent TB infection.

A New Age-Structured Multiscale Model of the Hepatitis C Virus Life-Cycle During Infection and Therapy With Direct-Acting Antiviral Agents

The dynamics of hepatitis C virus (HCV) RNA during translation and replication within infected cells were added to a previous age-structured multiscale mathematical model of HCV infection and treatment. The model allows the study of the dynamics of HCV RNA inside infected cells as well as the release of virus from infected cells and the dynamics of subsequent new cell infections. The model was used to fit in vitro data and estimate parameters characterizing HCV replication. This is the first model to our knowledge to consider both positive and negative strands of HCV RNA with an age-structured multiscale modeling approach. Using this model we also studied the effects of direct-acting antiviral agents (DAAs) in blocking HCV RNA intracellular replication and the release of new virions and fit the model to in vivo data obtained from HCV-infected subjects under therapy.

Mathematical model of the effects of government intervention and rehabilitation of prostitution

In the United States, prostitution is considered illegal in all but one state; Nevada allows some legal activities in exchange for substantial guidelines. In 2010, approximately 43,600 females were arrested for prostitution. Numerous intervention programs were established in order to obstruct the lifestyle of a prostitute (PRP, Project ROSE, etc.). There are many documentations and programs that share their forethought on prostitution; however, few target prostitution directly. To determine the dynamics of prostitution, this paper constructs a four-class compartmental model that focuses on the effectiveness of government intervention and rehabilitation of prostitutes mathematically. The basic reproductive number, [Formula: see text], helps to discover the threshold values for the dynamics of prostitution to become both prevalent or absent in society. This paper predominately observes government intervention to curtail a prostitution prevalent society. Various parameters and variables help to define and indicate the dynamics of prostitution to construct viable simulations. Successful prostitution interaction prevention deemed essential in prostitution prevention; however, government intervention corresponding with successful rehabilitation competitively challenges prostitution interaction prevention in reducing basic reproductive values.

Mathematical model and intervention strategies for mitigating tuberculosis in the Philippines

Tuberculosis (TB) is the sixth leading cause of morbidity and mortality in the Philippines. Although significant progress has been made in the detection and cure of TB under the Directly Observed Treatment Short Course, battling against the disease is still a burdensome task. It demands a concerted effort for specific and effective interventions. In this work, a mathematical TB model fitted to the Philippine data is developed to understand its transmission dynamics. Different control strategies such as distancing, latent case finding, case holding, active case finding controls, and combinations thereof are investigated within the framework of optimal control theory. This study proposes optimal control strategies for reducing the number of high-risk latent and infectious TB patients with minimum intervention implementation costs. Results suggest that distancing control is the most efficient control strategy when a single intervention is utilized. However, full scale employment of the distancing control measure is a daunting task. This burden can be circumvented by the combination of other control interventions. Our noble finding in this study is that enhancing active case finding control instead of case holding control together with distancing and latent case finding control is shown to have significant potential for curtailing the spread of TB in the Philippines.

Coupled, multi-strain epidemic models of mutating pathogens

We introduce and analyze coupled, multi-strain epidemic models designed to simulate the emergence and dissemination of mutant (e.g. drug-resistant) pathogen strains. In particular, we investigate the mathematical and biological properties of a general class of multi-strain epidemic models in which the infectious compartments of each strain are coupled together in a general manner. We derive explicit expressions for the basic reproduction number of each strain and highlight their importance in regulating the system dynamics (e.g. the potential for an epidemic outbreak) and the existence of nonnegative endemic solutions. Importantly, we find that the basic reproduction number of each strain is independent of the mutation rates between the strains - even under quite general assumptions for the form of the infectious compartment coupling. Moreover, we verify that the coupling term promotes strain coexistence (as an extension of the competitive exclusion principle) and demonstrate that the strain with the greatest reproductive capacity is not necessarily the most prevalent. Finally, we briefly discuss the implications of our results for public health policy and planning.

A model of population dynamics of TB in a prison system and application to South Africa

BACKGROUND

Tuberculosis (TB) continues to spread in South African prisons in particular, as prisons are over-capacitated and have poor ventilation. The awaiting trial detainees are not screened on admission and are at high risk of getting infected with TB.

RESULTS

We propose a compartmental model to describe the population dynamics of TB disease in prisons. Our model considers the inflow of susceptible, exposed and TB infectives into the prison population. Removal of individuals out of the prison population can be either by death or by being released from prison, as compared to a general population in which removal is only by death. We describe conditions, including non-inflow of infectives into the prison, which will ensure that TB can be eradicated from the prison population. The model is calibrated for the South African prison system, by using data in existing literature. The model can be used to make quantitative projections of TB prevalence and to measure the effect of interventions. Illustrative simulations in this regard are presented. The model can be used for other prison populations too, if data is available to calculate the model parameters.

CONCLUSIONS

Various simulations generated with our model serve to illustrate how it can be utilized in making future projections of the levels of prevalence of TB, and to quantify the effect of interventions such as screening, treatment or reduction of transmission parameter values through improved living conditions for inmates. This makes it particularly useful as there are various targets set by the World Health Organization and by governments, for reduction of TB prevalence and ultimately its eradication. Towards eradication of TB from a prison system, the theorem on global stability of the disease-free state is a useful indicator.

Optimally capturing latency dynamics in models of tuberculosis transmission

Although different structures are used in modern tuberculosis (TB) models to simulate TB latency, it remains unclear whether they are all capable of reproducing the particular activation dynamics empirically observed. We aimed to determine which of these structures replicate the dynamics of progression accurately. We reviewed 88 TB-modelling articles and classified them according to the latency structure employed. We then fitted these different models to the activation dynamics observed from 1352 infected contacts diagnosed in Victoria (Australia) and Amsterdam (Netherlands) to obtain parameter estimates. Six different model structures were identified, of which only those incorporating two latency compartments were capable of reproducing the activation dynamics empirically observed. We found important differences in parameter estimates by age. We also observed marked differences between our estimates and the parameter values used in many previous models. In particular, when two successive latency phases are considered, the first period should have a duration that is much shorter than that used in previous studies. In conclusion, structures incorporating two latency compartments and age-stratification should be employed to accurately replicate the dynamics of TB latency. We provide a catalogue of parameter values and an approach to parameter estimation from empiric data for calibration of future TB-models.

Mixed vaccination strategy for the control of tuberculosis: A case study in China

This study first presents a mathematical model of TB transmission considering BCG vaccination compartment to investigate the transmission dynamics nowadays. Based on data reported by the National Bureau of Statistics of China, the basic reproduction number is estimated approximately as R0=1.1892. To reach the new End TB goal raised by WHO in 2015, considering the health system in China, we design a mixed vaccination strategy. Theoretical analysis indicates that the infectious population asymptotically tends to zero with the new vaccination strategy which is the combination of constant vaccination and pulse vaccination. We obtain that the control of TB is quicker to achieve with the mixed vaccination. The new strategy can make the best of current constant vaccination, and the periodic routine health examination provides an operable environment for implementing pulse vaccination in China. Numerical simulations are provided to illustrate the theoretical results and help to design the final mixed vaccination strategy once the new vaccine comes out.

A patchy model for the transmission dynamics of tuberculosis in sub-Saharan Africa

Tuberculosis (TB) spreads through contact between a susceptible person and smear positive pulmonary TB case (TPM+). The spread of TB is highly dependent on people migration between cities or regions that may have different contact rates and different environmental parameters, leading to different disease spread speed in the population. In this work, a metapopulation model, i.e., networks of populations connected by migratory flows, which overcomes the assumption of homogeneous mixing between different regions was constructed. The TB model was combined to a simple demographic structure for the population living in a multi-patch environment (cities, towns, regions or countries). The model consist of a system of differential equations coupling TB epidemic at different strength and mobility between the patches. Constant recruitment rate, slow and fast progression to the disease, effective chemoprophylaxis, diagnostic and treatment are taken into account to make the model including the reality of people in the sub-Saharan African countries. The basic reproduction number ( R 0 ) was computed and it was demonstrated that the disease-free equilibrium is globally asymptotically stable ifR 0 < 1 . WhenR 0 > 1 , the disease-free equilibrium is unstable and there exists one endemic equilibrium. Moreover, the impact of increasing migration rate between patches on the TB spread was quantified using numerical implementation of the model. Using an example on 15 inter-connected patches on the same road, we demonstrated that most people was most likely to get infected if the disease starts in a patch in the middle than in border patches.

Optimal control of a tuberculosis model with state and control delays

We introduce delays in a tuberculosis (TB) model, representing the time delay on the diagnosis and commencement of treatment of individuals with active TB infection. The stability of the disease free and endemic equilibriums is investigated for any time delay. Corresponding optimal control problems, with time delays in both state and control variables, are formulated and studied. Although it is well-known that there is a delay between two to eight weeks between TB infection and reaction of body's immune system to tuberculin, delays for the active infected to be detected and treated, and delays on the treatment of persistent latent individuals due to clinical and patient reasons, which clearly justifies the introduction of time delays on state and control measures, our work seems to be the first to consider such time-delays for TB and apply time-delay optimal control to carry out the optimality analysis.

High rates of multidrug-resistant and rifampicin-resistant tuberculosis among re-treatment cases: where do they come from?

BACKGROUND

Globally 3.9% of new and 21% of re-treatment tuberculosis (TB) cases are multidrug-resistant or rifampicin-resistant (MDR/RR), which is often interpreted as evidence that drug resistance results mainly from poor treatment adherence. This study aims to assess the respective contributions of the different causal pathways leading to MDR/RR-TB at re-treatment.

METHODS

We use a simple mathematical model to simulate progression between the different stages of disease and treatment for patients diagnosed with TB. The model is parameterised using region and country-specific TB disease burden data reported by the World Health Organization (WHO). The contributions of four separate causal pathways to MDR/RR-TB among re-treatment cases are estimated: I) initial drug-susceptible TB with resistance amplification during treatment; II) initial MDR/RR-TB inappropriately treated as drug-susceptible TB; III) MDR/RR-TB relapse despite appropriate treatment; and IV) re-infection with MDR/RR-TB.

RESULTS

At the global level, Pathways I, II, III and IV contribute 38% (28-49, 95% Simulation Interval), 44% (36-52, 95% SI), 6% (5-7, 95% SI) and 12% (7-19, 95% SI) respectively to the burden of MDR/RR-TB among re-treatment cases. Pathway II is dominant in the Western Pacific (74%; 67-80 95% SI), Eastern Mediterranean (68%; 60-74 95% SI) and European (53%; 48-59 95% SI) regions, while Pathway I makes the greatest contribution in the American (53%; 40-66 95% SI), African (43%; 28-61 95% SI) and South-East Asian (50%; 40-59 95% SI) regions.

CONCLUSIONS

Globally, failure to diagnose MDR/RR-TB at first presentation is the leading cause of the high proportion of MDR/RR-TB among re-treatment cases. These findings highlight the need for contextualised solutions to limit the impact and spread of MDR/RR-TB.

The optimal treatment of an infectious disease with two strains

This paper explores the optimal treatment of an infectious disease in a Susceptible-Infected-Susceptible model, where there are two strains of the disease and one strain is more infectious than the other. The strains are perfectly distinguishable, instantly diagnosed and equally costly in terms of social welfare. Treatment is equally costly and effective for both strains. Eradication is not possible, and there is no superinfection. In this model, we characterise two types of fixed points: coexistence equilibria, where both strains prevail, and boundary equilibria, where one strain is asymptotically eradicated and the other prevails at a positive level. We derive regimes of feasibility that determine which equilibria are feasible for which parameter combinations. Numerically, we show that optimal policy exhibits switch points over time, and that the paths to coexistence equilibria exhibit spirals, suggesting that coexistence equilibria are never the end points of optimal paths.

Complex Dynamical Behaviour in an Epidemic Model with Control

We analyse the dynamical behaviour of a simple, widely used model that integrates epidemiological dynamics with disease control and economic constraint on the control resources. We consider both the deterministic model and its stochastic counterpart. Despite its simplicity, the model exhibits mathematically rich dynamics, including multiple stable fixed points and stable limit cycles arising from global bifurcations. We show that the existence of the limit cycles in the deterministic model has important consequences in modelling the range of potential effects the control can have. The stochastic effects further interact with the deterministic dynamical structure by facilitating transitions between different attractors of the system. The interaction is important for the predictive power of the model and in using the model to optimize allocation when resources for control are limited. We conclude that when studying models with constrained control, special care should be given to the dynamical behaviour of the system and its interplay with stochastic effects.

A two-strain TB model with multiple latent stages

A two-strain tuberculosis (TB) transmission model incorporating antibiotic-generated TB resistant strains and long and variable waiting periods within the latently infected class is introduced. The mathematical analysis is carried out when the waiting periods are modeled via parametrically friendly gamma distributions, a reasonable alternative to the use of exponential distributed waiting periods or to integral equations involving ``arbitrary'' distributions. The model supports a globally-asymptotically stable disease-free equilibrium when the reproduction number is less than one and an endemic equilibriums, shown to be locally asymptotically stable, or l.a.s., whenever the basic reproduction number is greater than one. Conditions for the existence and maintenance of TB resistant strains are discussed. The possibility of exogenous re-infection is added and shown to be capable of supporting multiple equilibria; a situation that increases the challenges faced by public health experts. We show that exogenous re-infection may help established resilient communities of actively-TB infected individuals that cannot be eliminated using approaches based exclusively on the ability to bring the control reproductive number just below 1.

Tuberculosis Epidemiology at the Country Scale: Self-Limiting Process and the HIV Effects

BACKGROUND

The global spread of the human immunodeficiency virus (HIV) is the main hypothesis behind tuberculosis (TB) positive trends in the last decades, according to modeling studies and World Health Organization Reports (WHO). On one hand, TB (WHO) reports do not explicitly consider a modeling approach, but cover country and global TB trends. On the other hand, modeling studies usually do not cover the scale of WHO reports, because of the amount of parameters estimated to describe TB natural history. Here we combined these two principal sources of TB studies covering TB High Burden Countries (HBCs) dynamics. Our main goals were: (i) to detect the endogenous component of TB dynamics since 1974 for TB HBCs; and (ii) to explore the HIV exogenous effects on TB models`parameters.

METHODS AND FINDINGS

We explored the relationship between the TB per capita population rate of change (RI) and the infectious class size following an endogenous/exogenous framework. RI can be affected by intra-population processes (i.e. competition, predation) and exogenous variables like HIV. We found that TB dynamics had always a strong endogenous component, represented by a negative correlation between TB population size and RI, which was captured by the discrete logistic model. Moreover, we explored the HIV exogenous effects on TB models`parameters. We found that overall the TB+HIV logistic model was more parsimonious than TB model alone, principally in the African region. Our results showed that HIV affected principally TB carrying capacity, as expected by the known HIV effects on TB natural-history. We also tested if DOTS (Directly Observed Treatment Short-Course Strategy), poverty levels and BCG (Bacillus Calmette-Guérin) coverage explained the models´ residuals variances, but they did not.

CONCLUSIONS

Since 1974, TB dynamics were categorized in distinct chronological domains, with different dynamics but nearly the same underlying mechanism: a negative relationship between RI and infected class size (i.e. self-limiting). In the last decades, not only HIV spread represented a new TB chronological domain, but it also has been pushing TB carrying capacity (K) to higher levels. TB has a complex natural-history and imposes real challenges to model its dynamics. Yet, we were able to explore and reveal the main drivers of TB dynamics for HBCs since 1974, through a simple approach. Based on our results, we suggest that the endogenous view should be considered as a plausible hypothesis to model and explain TB dynamics and that future World Health Organization reports could include the endogenous/exogenous framework as a supplement to help to decipher the main drivers of TB dynamics and other diseases.

Tuberculosis in Cape Town: An age-structured transmission model

BACKGROUND

Tuberculosis (TB) is the leading cause of death in South Africa. The burden of disease varies by age, with peaks in TB notification rates in the HIV-negative population at ages 0-5, 20-24, and 45-49 years. There is little variation between age groups in the rates in the HIV-positive population. The drivers of this age pattern remain unknown.

METHODS

We developed an age-structured simulation model of Mycobacterium tuberculosis (Mtb) transmission in Cape Town, South Africa. We considered five states of TB progression: susceptible, infected (latent TB), active TB, treated TB, and treatment default. Latently infected individuals could be re-infected; a previous Mtb infection slowed progression to active disease. We further considered three states of HIV progression: HIV negative, HIV positive, on antiretroviral therapy. To parameterize the model, we analysed treatment outcomes from the Cape Town electronic TB register, social mixing patterns from a Cape Town community and used literature estimates for other parameters. To investigate the main drivers behind the age patterns, we conducted sensitivity analyses on all parameters related to the age structure.

RESULTS

The model replicated the age patterns in HIV-negative TB notification rates of Cape Town in 2009. Simulated TB notification rate in HIV-negative patients was 1000/100,000 person-years (pyrs) in children aged <5 years and decreased to 51/100,000 in children 5-15 years. The peak in early adulthood occurred at 25-29 years (463/100,000 pyrs). After a subsequent decline, simulated TB notification rates gradually increased from the age of 30 years. Sensitivity analyses showed that the dip after the early adult peak was due to the protective effect of latent TB and that retreatment TB was mainly responsible for the rise in TB notification rates from the age of 30 years.

CONCLUSION

The protective effect of a first latent infection on subsequent infections and the faster progression in previously treated patients are the key determinants of the age-structure of TB notification rates in Cape Town.