Parameter Estimation and Optimal Control of the Dynamics Of Transmission of Tuberculosis with Application to Cameroon

This paper deals with the problem of parameter estimation and optimal control of a tuberculosis (TB) model with seasonal fluctuations. We first present a uncontrolled TB model with seasonal fluctuations. We present the theoretical analysis of the uncontrolled TB model without seasonal fluctuations. After, we propose a numerical study to estimate the unknown parameters of the TB model with seasonal fluctuations according to demographic and epidemiological data from Cameroon. Simulation results are in good accordance with the seasonal variation of the new active reported cases of TB in Cameroon. Using this TB model with seasonality, the tuberculosis control is formulated and solved as an optimal control problem, indicating how control terms on the chemoprophylaxis and treatment should be introduced in the considered TB model to reduce the number of individuals with active TB. Results provide a framework for designing cost-effective strategies for TB with two strategies of intervention

Backward bifurcation and hysteresis in models of recurrent tuberculosis

An epidemiological model is presented that provides a comprehensive description of the transmission pathways involved for recurrent tuberculosis (TB), whereby cured individuals can become reinfected. Our main goal is to determine conditions that lead to the appearance of a backward bifurcation. This occurs when an asymptotically stable infection free equilibrium concurrently exists with a stable non-trivial equilibria even though the basic reproduction number R 0 is less than unity. Although, some 10-30% cases of TB are recurrent, the role of recurrent TB as far as the formation of backward bifurcation is concerned, is rarely if ever studied. The model used here incorporates progressive primary infection, exogenous reinfection, endogenous reactivation and recurrent TB as transmission mechanisms that contribute to TB progression. Unlike other studies of TB dynamics that make use of frequency dependent transmission rates, our analysis provides exact backward bifurcation threshold conditions without resorting to commonly applied approximations and simplifying assumptions. Exploration of the model through analytical and numerical analysis reveal that recurrent TB is sometimes capable of triggering hysteresis effects which allow TB to persist when R 0 < 1 even though there is no backward bifurcation. Furthermore, recurrent TB can independently induce backward bifurcation phenomena if it exceeds a certain threshold

A minimal model coupling communicable and non-communicable diseases

This work presents a model combining the simplest communicable and non-communicable disease models. The latter is, by far, the leading cause of sickness and death in the World, and introduces basal heterogeneity in populations where communicable diseases evolve. The model can be interpreted as a risk-structured model, another way of accounting for population heterogeneity. Our results show that considering the non-communicable disease (in the end, heterogeneous populations) allows the communicable disease to become endemic even if the basic reproduction number is less than $1$. This feature is known as subcritical bifurcation. Furthermore, ignoring the non-communicable disease dynamics results in overestimating the reproduction number and, thus, giving wrong information about the actual number of infected individuals. We calculate sensitivity indices and derive interesting epidemic-control information.Comment: 19 pages, 5 figure

A minimal model coupling communicable and non-communicable diseases

This work presents a model combining the simplest communicable and non-communicable disease models. The latter is, by far, the leading cause of sickness and death in the World, and introduces basal heterogeneity in populations where communicable diseases evolve. The model can be interpreted as a risk-structured model, another way of accounting for population heterogeneity. Our results show that considering the non-communicable disease (in the end, a dynamic heterogeneous population) allows the communicable disease to become endemic even if the basic reproduction number is less than 1. This feature is known as subcritical bifurcation. Furthermore, ignoring the non-communicable disease dynamics results in overestimating the basic reproduction number and, thus, giving wrong information about the actual number of infected individuals. We calculate sensitivity indices and derive interesting epidemic-control information.Agencia Estatal de InvestigaciónMinisterio de Economía y Competitivida

Backward bifurcation and reinfection in mathematical models of tuberculosis

Mathematical models are widely used for understanding the transmission mechanisms and control of infectious diseases. Numerous infectious diseases such as those caused by bacterial and viral infections do not confer life long immunity after recovering from the first episode. Consequently, they are characterized by partial or complete loss of immunity and subsequent reinfection. This thesis explores the epidemiological implications of loss of immunity using simple and complex mathematical models. First, a simple basic model mimicking transmission mechanisms of tuberculosis (TB) is proposed with the aim of correcting problems that are often repeated by mathematical modellers when determining underlying bifurcation structures. Specifically, the model makes transparent the problems that may arise if one aggregates all the bifurcation parameters when computing backward bifurcation thresholds and structures. The backward bifurcation phenomenon is an important concept for public health and disease management. This is because backward bifurcation signals that disease will not be eliminated even when the basic reproduction number R0 is decreased below unity; rather, for the disease to be eliminated, R0 has to be reduced below another critical threshold. I provide conditions to find the threshold correctly. Secondly, the simple basic TB model is extended to incorporate epidemiological and biological aspects pertinent to TB transmission such as recurrent TB, which is defined as a second episode of TB following successful recovery from a previous episode. I study the conditions for backward bifurcation in this extended model that features recurrent TB. Mathematical techniques based on the center manifold approach, are used to derive an exact backward bifurcation threshold. Furthermore, both analytical and numerical findings reveal that recurrent TB is capable of inducing a new and rare hysteresis effect where TB will persist when the basic reproduction number is below unity even though there is no backward bifurcation. Moreover, when the reinfection pathway among latently infected individuals is switched off, leaving only recurrent TB, the model analysis indicates that recurrent TB can independently induce a backward bifurcation. However, this will only occur if recurrent TB transmission exceeds a certain threshold. Although this threshold seems to be relatively high when realistic parameters are used, it falls within the recent range estimated in the relevant literature. The second TB model is extended by dividing the latent compartment into two: fast (early latent) and slow (late latent) latent compartments, to enhance realism. Individuals in both early and late compartments are subjected to treatment. The proposed TB model is used to investigate how heterogeneity in host susceptibility influences the effectiveness of treatment. It is found that making the assumption that individuals treated with preventive therapy and recovered individuals (previously treated for active TB) acquire equal levels of protection after initial infection, and are therefore reinfected at the same rate, may obscure dynamics that are imperative when designing intervention strategies. Comparison of reinfection rates between cohorts treated with preventive therapy and recovered individuals who were previously treated from active TB provides important epidemiological insights. That is, the reinfection parameter accounting for the relative rate of reinfection of the cohort treated with preventive therapy is the one that plays the key role in generating qualitative changes in TB dynamics. In contrast, the parameter accounting for the risk of reinfection among recovered individuals (previously treated for active TB) does not play a significant role. The study shows that preventive treatment during early latency is always beneficial regardless of the level of susceptibility to reinfection. And 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: (a) for (very) low burden settings, the effect on reactivation predominates and burden declines; (b) for high burden settings, the effect on reinfection predominates and burden increases. This is mostly observed 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. Finally, a mathematical model that examines how heroin addiction spreads in society is formulated. The model has many commonalities with the TB model. The global stability properties of the proposed model are analysed using both the Lyapunov direct method and the geometric approach by Li and Muldowney. It is shown that even for a four dimensional model, the use of two well known nonlinear stability techniques becomes nontrivial. When all the parameters of the model are accounted for, it is difficult if not impossible, to design a Lyapunov function. Here I apply the geometric approach to establish a global condition that accounts for all model parameters. If the condition is satisfied, then heroin persistence within the community is globally stable. However, if the global condition is not satisfied heroin users can oscillate periodically in number. Numerical simulations are also presented to give a more complete representation of the model dynamics. Sensitivity analysis performed by Latin hypercube sampling (LHS) suggests that the effective contact rate in the population, the relapse rate of heroin users undergoing treatment, and the extent of saturation of heroin users, are the key mechanisms fuelling heroin epidemic proliferation. However, in the long term, relapse of heroin users undergoing treatment back to a heroin using career, has the most significant impact

Modeling tuberculosis:a compromise between biological realism and mathematical tractability

Tese de doutoramento, Matemática (Análise Matemática), 2009, Universidade de Lisboa, Faculdade de CiênciasDisponível no document

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

Mathematical modelling and analysis of HIV transmission dynamics

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis firstly presents a nonlinear extended deterministic Susceptible-Infected (SI) model for assessing the impact of public health education campaign on curtailing the spread of the HIV pandemic in a population. Rigorous qualitative analysis of the model reveals that, in contrast to the model without education, the full model with education exhibits the phenomenon of backward bifurcation (BB), where a stable disease-free equilibrium coexists with a stable endemic equilibrium when a certain threshold quantity, known as the effective reproduction number (Reff ), is less than unity. Furthermore, an explicit threshold value is derived above which such an education campaign could lead to detrimental outcome (increase disease burden), and below which it would have positive population-level impact (reduce disease burden in the community). It is shown that the BB phenomenon is caused by imperfect efficacy of the public health education program. The model is used to assess the potential impact of some targeted public health education campaigns using data from numerous countries. The second problem considered is a Susceptible-Infected-Removed (SIR) model with two types of nonlinear treatment rates: (i) piecewise linear treatment rate with saturation effect, (ii) piecewise constant treatment rate with a jump (Heaviside function). For Case (i), we construct travelling front solutions whose profiles are heteroclinic orbits which connect either the disease-free state to an infected state or two endemic states with each other. For Case (ii), it is shown that the profile has the following properties: the number of susceptible individuals is monotone increasing and the number of infectives approaches zero, while their product converges to a constant. Numerical simulations are shown which confirm these analytical results. Abnormal behavior like travelling waves with non-monotone profile or oscillations are observed.Kano State Government of Nigeri

Mathematical model of HIV/AIDS, tuberculosis and their coevolution with optimal control: A case study in Ethiopia

The communicable disease tuberculosis (TB), human immunodeficiency virus/acquired immune deficiency syndrome (HIV/AIDS) disease, and their co-infection are the most serious public health issues in the world. In this thesis, three population level mathematical models of the three infections min Ethiopia are developed and analyzed. The first model considers the dynamics of HIV/AIDS, which comprise the following exclusive classes of individuals, the aware and unaware susceptibles, undiagnosed HIV infectious, diagnosed HIV infectious with and without AIDS symptoms, and those under HIV treatment. This model considers the rate of becoming aware and unaware as a function of media campaigns, whereas screening and treatment rates are constant. The effective reproduction number, equilibria, and nature of stability were formulated. The bifurcation occurs when the effective reproduction number is equal to unity. This model is extended to a new model which incorporates interventions such as preventive, screening, and treatment strategies. In this model, the optimal control problem is formulated and solved analytically. In addition to this, the optimality system is derived and solved numerically using the forward-backward sweep method (FBSM). Finally, the cost-effectiveness of some combined control strategies is derived. The second model reflects the TB transmission dynamics with drug resistance TB (DR-TB). The two infectious TB stages, namely drug-sensitive TB and drug-resistant TB, are considered in the model. Assuming that drug-sensitive TB can be cured by first-line anti-TB drugs. In fact, once the Tubercle Bacilli become resistant to one or more anti-TB drugs, the drug-resistance TB occurs. The model is analyzed analytically and extended to an optimal control problem via incorporating preventive efforts, case finding, and case holding. In the study, four different strategies are introduced based on different combination of measures. The optimal control problem is examined both analytically and numerically. The third model describes a new mathematical model of human immunodeficiency virus (HIV) associated with tuberculosis (TB). This full TB-HIV co-infection model is analyzed analytically. Which is extended to an optimal control problem by using controlling variables such as preventive efforts, case finding effort for TB, and HIV treatment. We proposed four strategies, which are combinations of two or more control measures at a time. The model with controls is analyzed both analytically and numerically. The numerical results are derived using the classical Runge-Kutta method of order four (RK4-method). The finding suggests that optimal combination strategies are used to reduce both the disease burden and the cost of intervention. Further, the cost- effectiveness of each strategy is assessed to identify the best cost-effective approach the fight against TB-HIV co-infection in Ethiopia.Mathematical SciencesD. Phil. (Applied Mathematics

Understanding the Impact of Social Factors on the Transmission Dynamics of Infectious Diseases Across Highly Heterogeneous Risk Environments.

abstract: This dissertation explores the impact of environmental dependent risk on disease dynamics within a Lagrangian modeling perspective; where the identity (defined by place of residency) of individuals is preserved throughout the epidemic process. In Chapter Three, the impact of individuals who refuse to be vaccinated is explored. MMR vaccination and birth rate data from the State of California are used to determine the impact of the anti-vaccine movement on the dynamics of growth of the anti-vaccine sub-population. Dissertation results suggest that under realistic California social dynamics scenarios, it is not possible to revert the influence of anti-vaccine contagion. In Chapter Four, the dynamics of Zika virus are explored in two highly distinct idealized environments defined by a parameter that models highly distinctive levels of risk, the result of vector and host density and vector control measures. The underlying assumption is that these two communities are intimately connected due to economics with the impact of various patterns of mobility being incorporated via the use of residency times. In short, a highly heterogeneous community is defined by its risk of acquiring a Zika infection within one of two "spaces," one lacking access to health services or effective vector control policies (lack of resources or ignored due to high levels of crime, or poverty, or both). Low risk regions are defined as those with access to solid health facilities and where vector control measures are implemented routinely. It was found that the better connected these communities are, the existence of communities where mobility between risk regions is not hampered, lower the overall, two patch Zika prevalence. Chapter Five focuses on the dynamics of tuberculosis (TB), a communicable disease, also on an idealized high-low risk set up. The impact of mobility within these two highly distinct TB-risk environments on the dynamics and control of this disease is systematically explored. It is found that collaboration and mobility, under some circumstances, can reduce the overall TB burden.Dissertation/ThesisDoctoral Dissertation Applied Mathematics for the Life and Social Sciences 201