Modelling the effect of tuberculosis on the spread of HIV infection in a population with density-dependent birth and death rate

A nonlinear mathematical model is proposed to study the effect of tuberculosis on the spread of HIV infection in a logistically growing human population. The host population is divided into four sub classes of susceptibles, TB infectives, HIV infectives (with or without TB) and that of AIDS patients. The model exhibits four equilibria namely, a disease free, HIV free, TB free and an endemic equilibrium. The model has been studied qualitatively using stability theory of nonlinear differential equations and computer simulation. We have found a threshold parameter R₀ which is if less than one, the disease free equilibrium is locally asymptotically stable otherwise for R₀>1, at least one of the infections will be present in the population. It is shown that the positive endemic equilibrium is always locally stable but it may become globally stable under certain conditions showing that the disease becomes endemic. It is found that as the number of TB infectives decreases due to recovery, the number of HIV infectives also decreases and endemic equilibrium tends to TB free equilibrium. It is also observed that number of AIDS individuals decreases if TB is not associated with HIV infection. A numerical study of the model is also performed to investigate the influence of certain key parameters on the spread of the disease.     

The effect of immigrant communities coming from higher incidence tuberculosis regions to a host country

We introduce a new tuberculosis (TB) mathematical model, with 25 state-space variables where 15 are evolution disease states (EDSs), which generalises previous models and takes into account the (seasonal) flux of populations between a high incidence TB country (A) and a host country (B) with low TB incidence, where (B) is divided into a community (G) with high percentage of people from (A) plus the rest of the population (C). Contrary to some beliefs, related to the fact that agglomerations of individuals increase proportionally to the disease spread, analysis of the model shows that the existence of semi-closed communities are beneficial for the TB control from a global viewpoint. The model and techniques proposed are applied to a case-study with concrete parameters, which model the situation of Angola (A) and Portugal (B), in order to show its relevance and meaningfulness. Simulations show that variations of the transmission coefficient on the origin country has a big influence on the number of infected (and infectious) individuals on the community and the host country. Moreover, there is an optimal ratio for the distribution of individuals in (C) versus (G), which minimizes the reproduction number \(R_0\). Such value does not give the minimal total number of infected individuals in all (B), since such is attained when the community (G) is completely isolated (theoretical scenario). Sensitivity analysis and curve fitting on \(R_0\) and on EDSs are pursuit in order to understand the TB effects in the global statistics, by measuring the variability of the relevant parameters. We also show that the TB transmission rate \(\beta \) does not act linearly on \(R_0\), as it is common in compartment models where system feedback or group interaction do not occur. Further, we find the most important parameters for the increase of each EDS.     

Mathematical model on pulmonary and multidrug-resistant tuberculosis patients with vaccination

Tuberculosis is a global epidemic disease and almost two billion people across the globe are infected with the tuberculosis bacilli. Many countries like China, Europe and United States has achieved dramatic decrease in TB mortality rate but country like India is still struggling hard to control this epidemic. Jharkhand one of the states of India is highly epidemic toward this disease. We propose a mathematical model to understand the spread of tuberculosis disease in human population for both pulmonary and drug-resistant subjects. A number of new vaccines are currently in development. Keeping in mind, vaccination as one of the treatment for TB patients may be infant or adult in future; an assumption for the transfer of proportion of susceptible population to the vaccination class is considered. Quarantine class is also considered in our epidemic model for multidrug-resistant patients, and it is observed that it may play a vital role for controlling the disease. Threshold and equilibria are obtained and the condition for epidemic under different conditions of threshold is established. Real parametric values of the Jharkhand state are taken into account to simulate the system developed, and the results so obtained validate our analytical results.     

System Dynamic simulation of treatment policies to address colliding epidemics of tuberculosis,drug resistant tuberculosis and injecting drug users driven HIV in Russia

The explosive increase in the number of people infected with tuberculosis (TB),multi drug resistant tuberculosis (MDRTB), and injecting drug users (IDU)HIV/AIDS has become a serious public health challenge in Russia. The WorldHealth Organization is recommending policies including simultaneous use ofhighly active antiretroviral therapy (HAART) to treat HIV/AIDS and second linedrugs to treat MDRTB. We developed a System Dynamics simulation model torepresent the dynamic transmission of TB, MDRTB and human immunodeficiency virus(HIV). The model simulated scenarios regarding MDRTB cure rate and HAARTcoverage, that is the HIV/AIDS population covered by HAART. The results over a20-year period indicate that reduction in TB and HIV-associated TB deaths wouldbe negligible for HAART coverage up to 50%. The reduction is onlysignificant for HAART coverage of 70% and above. Similarly, high MDRTBcure rate reduces significantly deaths from TB and MDRTB and this reduction ismore important as the HAART coverage is increased.     

Global stability of two models with incomplete treatment for tuberculosis

Two tuberculosis (TB) models with incomplete treatment are investigated. It is assumed that the treated individuals may enter either the latent compartment due to the remainder of Mycobacterium tuberculosis or the infectious compartment due to the treatment failure. The first model is a simple one with treatment failure reflecting the current TB treatment fact in most countries with high tuberculosis incidence. The second model refines the simple one by dividing the latent compartment into slow and fast two kinds of progresses. This improvement can be used to describe the case that the latent TB individuals have been infected with some other chronic diseases (such as HIV and diabetes) which may weaken the immunity of infected individuals and shorten the latent period of TB. Both of the two models assume mass action incidence and exponential distributions of transfers between different compartments. The basic reproduction numbers of the two models are derived and their intuitive epidemiological interpretations are given. The global dynamics of two models are all proved by using Liapunov functions. At last, some strategies to control the spread of tuberculosis are discussed.     

Assessing alternative control strategies for systems with asymptotically stable equilibrium positions

In a number of optimal control applications, it is possible to arrange control guided only by an analysis of a system’s dynamic properties. These controls are customarily referred to as alternatives to those that satisfy the Pontryagin maximum principle. This work considers autonomous systems of ordinary differential equations with a terminal objective functional that at each fixed value of the control parameter have unique and asymptotically stable equilibrium positions. It is shown that the problem of arranging alternative control can then be reduced to a finite-dimensional problem of mathematical programming. An estimate of the alternative control error in terms of the objective functional is obtained. Sufficient conditions for obtaining this estimate are given. A mathematical model of leukemia therapy is considered as an example.     

The dynamics of an age-structured TB transmission model with relapse

This paper deals with the global dynamics for a tuberculosis transmission model with age-structure and relapse. The time delay in the progression from the latent individuals to becoming the infectious individuals is also considered in our model. We perform some rigorous analyses for the model, including presenting an explicit formula for the basic reproduction number of the model, addressing the persistence of the solution semiflow and the existence of a global attractor. Based on these analyses, we establish some results about stability and instability of the solutions for our model. At end, the model is applied to describe tuberculosis transmission in China. The number of the total population and the number of the annual newly reported TB cases both match the statistical data well. The number of the total population, the latent individuals, the infectious individuals, the Purified Protein Derivative (PPD) positive rate, and the prevalence rate from 2020 to 2035 all are presented.     

Impact of farming awareness based roguing,insecticide spraying and optimal control on the dynamics of mosaic disease

Control interventions and farming knowledge are equally important for plant disease control. In this article, a mathematical model has been derived using saturated response functions (nonlinear infection rate) for studying the dynamics of mosaic disease with farming awareness based roguing (removal of infected plants) and insecticide spraying . It is assumed that the use of roguing and spraying depend on the level of awareness about the disease. The model possesses three equilibria namely the trivial, which is always unstable, the disease-free equilibrium which is stable if the basic reproduction number is below unity and the coexisting which may be stable or can exhibit Hopf-bifurcation under certain condition. Finally, we have opted an optimal control problem introducing three control parameters for determining the optimal level of roguing, spraying and cost regarding media awareness for cost-effective control of mosaic disease. Numerical simulations establish the main results suggesting that the awareness campaigns through radio, TV advertisement are important for eradication of the disease. Also, awareness campaign, roguing and spraying should be incorporated with optimal level for cost effective control of mosaic disease.

     

Optimal control of time discrete two-phase flow governed by a diffuse interface model

We present results on optimal control of two-phase flows. The fluid is modeled by a thermodynamically consistent diffuse interface model and allows to treat fluids of different densities and viscosities. In earlier work we proposed an energy stable time discretization for this model that we now employ to derive existence of optimal controls for a time discrete optimal control problem. The control aim is to obtain a desired distribution of the two phases in the system. For this we investigate three control actions. We use tangential Dirichlet boundary control and distributed control. We further consider the inverse problem of finding an initial distribution such that the evolution over a given time horizon starting from this value is close to a desired distribution. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A general system theory approach to rail freight car fleet sizing

The transportation of goods from shippers to consignees is a railroad's major activity. Rail freight cars are enormously expensive and a rail vehicle fleet represents one of the largest capital resources of most railroads. Resource allocation to rail freight cars is an extraordinary complex managerial problem. This paper describes the determination of an optimal number of rail freight cars so as to satisfy the demand, on one hand, and minimize the total cost, on the other. A new mathematical model relying on optimal control theory is developed. The problem is formulated as the problem of finding an optimal regulator for a linear system, excited by Gaussian white noise, a quadratic performance index, and random initial conditions. The model has been tested on numerical examples.     

Effects of system parameters on the optimal policy structure in a class of queueing control problems

This paper studies a class of queueing control problems involving commonly used control mechanisms such as admission control and pricing. It is well established that in a number of these problems, there is an optimal policy that can be described by a few parameters. From a design point of view, it is useful to understand how such an optimal policy varies with changes in system parameters. We present a general framework to investigate the policy implications of the changes in system parameters by using event-based dynamic programming. In this framework, the control model is represented by a number of common operators, and the effect of system parameters on the structured optimal policy is analyzed for each individual operator. Whenever a queueing control problem can be modeled by these operators, the effects of system parameters on the optimal policy follow from this analysis.      

Computational Control of an HIV Model

We consider a computational approach to solving an optimal control formulation of optimal drug scheduling in HIV infected individuals. The optimal control problem is transformed using the control parameterisation enhancing technique (CPET), which enables efficient computation of an optimal control using a relatively coarse discretisation. A number of numerical difficulties with the model are discussed, and for illustration, numerical examples are solved.     

An optimal control approach to the design of periodic orbits for mechanical systems with impacts

In this paper we study the problem of designing periodic orbits for a special class of hybrid systems, namely mechanical systems with underactuated continuous dynamics and impulse events. We approach the problem by means of optimal control. Specifically, we design an optimal control based strategy that combines trajectory optimization, dynamics embedding, optimal control relaxation and root finding techniques. The proposed strategy allows us to design, in a numerically stable manner, trajectories that optimize a desired cost and satisfy boundary state constraints consistent with a periodic orbit. To show the effectiveness of the proposed strategy, we perform numerical computations on a compass biped model with torso.     

Dynamics and control of a system of two non‐interacting preys with common predator

In this paper, we consider a Holling type model, which describes the interaction between two preys with a common predator. First, we give some sufficient conditions for the globally asymptotic stability and prove that local stability implies global stability. Then, we present a set of sufficient conditions for the existence of a positive periodic solution with strictly positive components. Finally, the optimal control strategy is developed to minimize the number of predator and maximize the number of preys. We also show the existence of an optimal control for the optimal control problem and derive the optimality system. The technical tool used to determine the optimal strategy is the Pontryagin Maximum Principle. Finally, the numerical simulations of global stability and the optimal problem are given as the conclusion of this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability analysis and optimal control of a hand-foot-mouth disease (HFMD) model

In this paper, we propose a system of ordinary differential equations to model the hand-foot-mouth disease (HFMD). We derive the expression of the basic reproduction number . When , the system only has the disease free equilibrium, which is globally asymptotically stable; otherwise, the system is persistent. By sensitivity analysis, we identify the control parameters. Then we formulate an optimal control problem to find the optimal control strategy. These results are applied to the spread of HFMD in Mainland China. The basic reproduction number tells us that it is outbreak in China.     

由常规故障和临界人为错误引起系统故障的可修复系统稳态解的最优控制

针对由由常规故障和临界人为错误引起系统故障的可修复系统的模型,以范数指标泛函作为衡量系统可控性的标准,利用Banach空间理论讨论系统稳态解达到预期概率分布的最优控制问题,给出了其最优解存在唯一性.     

Mathematical analysis of a three-strain tuberculosis transmission model

In recent years, bacteria have become resistant to antibiotics, leading to a decline in the effectiveness of antibiotics in treating infectious diseases. A mathematical model for multi-strain tuberculosis transmission dynamics to assess the burden of drug-sensitive, multidrug-resistant and extensively drug-resistant tuberculosis is formulated and analyzed. Each single strain submodel is shown to exhibit backward bifurcation when the threshold parameter is less than unity. Both analytical and numerical results show that resistance to drugs increase with increase in drug use, that is, active tuberculosis treatment results in a reduction of drug sensitive and increase in multidrug-resistant tuberculosis. Furthermore, use of second line drugs results in a decrease of the multidrug-resistant and increase of extensively drug resistant tuberculosis as most cases of multidrug resistant tuberculosis occur as a result of inappropriate, misuse or mismanaged treatment. Both the analytic results and numerical simulations suggest that quarantine of extensively drug resistant TB cases in addition to treatment of other forms of TB may be able to reduce the spread of the epidemic in poor resource-settings.     

Instability of optimal equilibria in the minimum mass design of uniform shallow arches

Necessary and sufficient conditions for the minimum mass design of arbitrarily loaded uniform shallow arches are derived. The problem is posed as an optimal control problem with mass as the criterion, initial curvature and axial load as design variables, and with the differential equations of axial and transverse equilibrium of the arch as side conditions. Thus, an optimal equilibrium is associated with each optimal design, and the stability of these equilibria becomes an integral part of the problem solution. As an example, the design process is carried out for the sinusoidally loaded hinged-hinged arch with a fixed span. It turns out that, depending on the given load amplitude, the optimal equilibrium can be unstable, stable after snap-through, and nonunique with one equilibrium unstable and the other stable after snap-through, at the design load of the arch. In addition, a necessary condition for a local minimum is the same as the usual critical point condition in stability analysis, thus assuring the instability of the arch at the optimum. A brief survey of earlier work on the optimal design of arches and curved beams is also included.     

Higher order optimization and adaptive numerical solution for optimal control of monodomain equations in cardiac electrophysiology

In this work adaptive and high resolution numerical discretization techniques are demonstrated for solving optimal control of the monodomain equations in cardiac electrophysiology. A monodomain model, which is a well established model for describing the wave propagation of the action potential in the cardiac tissue, will be employed for the numerical experiments. The optimal control problem is considered as a PDE constrained optimization problem. We present an optimal control formulation for the monodomain equations with an extra-cellular current as the control variable which must be determined in such a way that excitations of the transmembrane voltage are damped in an optimal manner.The focus of this work is on the development and implementation of an efficient numerical technique to solve an optimal control problem related to a reaction-diffusions system arising in cardiac electrophysiology. Specifically a Newton-type method for the monodomain model is developed. The numerical treatment is enhanced by using a second order time stepping method and adaptive grid refinement techniques. The numerical results clearly show that super-linear convergence is achieved in practice.