Controlling the number of HIV infectives in a mobile population

The spread of the human immunodeficiency virus (HIV) depends prominently on the migration of people between different regions. An important consequence of this population mobility is that HIV control strategies that are optimal in a regional sense may not be optimal in a national sense. We formulate various mathematical control problems for HIV spread in mobile heterosexual populations, and show how optimal regional control strategies can be obtained that minimize the national spread of HIV. We apply the cross-entropy method to solve these highly multi-modal and non-linear optimization problems. We demonstrate the effectiveness of the method via a range of experiments and illustrate how the form of the optimal control function depends on the mathematical model used for the HIV spread.     

Chemotherapy for tumors: an analysis of the dynamics and a study of quadratic and linear optimal controls

We investigate a mathematical model of tumor-immune interactions with chemotherapy, and strategies for optimally administering treatment. In this paper we analyze the dynamics of this model, characterize the optimal controls related to drug therapy, and discuss numerical results of the optimal strategies. The form of the model allows us to test and compare various optimal control strategies, including a quadratic control, a linear control, and a state-constraint. We establish the existence of the optimal control, and solve for the control in both the quadratic and linear case. In the linear control case, we show that we cannot rule out the possibility of a singular control. An interesting aspect of this paper is that we provide a graphical representation of regions on which the singular control is optimal.     

Optimal population stabilization and control using the Leslie matrix model

We consider the problem of optimal stabilization and control of populations which follow the Leslie model dynamics, within state space and control systems theory and methodology. Various types of culling strategies are formulated and introduced into the Leslie model as control inputs, and their effect on global asymptotic stability is investigated. Our new approach provides answers to several unexplored problems. We show that in general it is possible to achieve a desired stable equilibrium population level, through the design of a class ofshifted-proportional stabilizing culling policies. Further, we formulate general non-linear constrained opitmization problems, for obtaining the cost-optimal policy among this generally infinite class of such stabilizing policies. The theoretical findings are illustrated through the solution of the problem over an infinite planning horizon for a numerical example. A comparative study of the costs and dynamic effects of various culling strategies also supports the mathematical results.     

Cancer immunotherapy, mathematical modeling and optimal control

Clinical immunologists, among other problems, routinely face a question: what is the best time and dose for a certain therapeutic agent to be administered to the patient in order to decrease/eradicate the pathological condition? In cancer immunotherapies the therapeutic agent is something able to elicit an immune response against cancer. The immune response has its own dynamics that depends on the immunogenicity of the therapeutic agent and on the duration of the immune response. The question then is "how can we decide when and how much of the drug to inject so to have a prolonged and effective immune response to the cancer?". This question can be addressed in mathematical terms in two stages: first one construct a mathematical model describing the cancer-immune interaction and secondly one applies the theory of optimal control to determine when and to which extent to stimulate the immune system by means of an immunotherapeutic agent administered in discrete variable doses within the therapeutic period. The solution of this mathematical problem is described and discussed in this article. We show that the method employed can be applied to find the optimal protocol in a variety of clinical problems where the kinetics of the drug or treatment and its influence on the physiologic/pathologic functions have been described by a system of ordinary differential equations.     

Impact of spatial heterogeneity on a predator-prey system dynamics

This paper deals with the study of a predator-prey model in a patchy environment. Prey individuals moves on two patches, one is a refuge and the second one contains predator individuals. The movements are assumed to be faster than growth and predator-prey interaction processes. Each patch is assumed to be homogeneous. The spatial heterogeneity is obtained by assuming that the demographic parameters (growth rates, predation rates and mortality rates) depend on the patches. On the predation patch, we use a Lotka-Volterra model. Since the movements are faster that the other processes, we may assume that the frequency of prey and predators become constant and we would get a global predator-prey model, which is shown to be a Lotka-Volterra one. However, this simplified model at the population level does not match the dynamics obtained with the complete initial model. We explain this phenomenom and we continue the analysis in order to give a two-dimensional predator-prey model that gives the same dynamics as that provided by the complete initial one. We use this simplified model to study the impact of spatial heterogeneity and movements on the system stability. This analysis shows that there is a globally asymptotically stable equilibrium in the positive quadrant, i.e. the spatial heterogeneity stabilizes the equilibrium.     

Stability analysis and optimal vaccination of an SIR epidemic model

Almost all mathematical models of diseases start from the same basic premise: the population can be subdivided into a set of distinct classes dependent upon experience with respect to the relevant disease. Most of these models classify individuals as either a susceptible individual S, infected individual I or recovered individual R. This is called the susceptible-infected-recovered (SIR) model. In this paper, we describe an SIR epidemic model with three components; S, I and R. We describe our study of stability analysis theory to find the equilibria for the model. Next in order to achieve control of the disease, we consider a control problem relative to the SIR model. A percentage of the susceptible populations is vaccinated in this model. We show that an optimal control exists for the control problem and describe numerical simulations using the Runge-Kutta fourth order procedure. Finally, we describe a real example showing the efficiency of this optimal control.     

A tunable multivariable nonlinear robust observer for biological systems

This paper presents a robust nonlinear asymptotic observer with adjustable convergence rate with a great potential of applicability for biological systems in which the main state variables are difficult and expensive to measure or such measurements do not exist. This observer scheme is based on the classical asymptotic observer, which is modified to allow the tuning of the convergence rate. It is shown that the proposed observer provides fast and satisfactory estimates when facing load disturbances, system failures and parameter uncertainty while maintaining the excellent robustness and stability properties of the classical asymptotic observer. The implementation of the tunable observer is carried out by numerical simulations of a mathematical model of an anaerobic digestion process used for wastewater treatment. The key results are examined and further developed.     

一类新Willis环脑动脉瘤系统的构造及最优控制

本文基于6-氨基乙酸减小血流速度及缓冲血流波动的性质,在6-氨基乙酸的作用下,构建了一个新的脑动脉瘤数学模型.该模型很好地体现了6-氨基乙酸、血流速度及血流速度变化率三者之间的相互作用关系.鉴于脑动脉瘤的医疗费用颇高及破裂后的死亡率较高,从而有必要从数学的角度研究该模型的最优控制,以致在一定条件下花费最小且医疗效果最佳.本文首先证明了该模型最优控制的存在性;其次通过构造Lagrangian函数及运用最大值原理,证明了最优控制的唯一性.从理论上得到Willis环脑动脉瘤内血流波动最小的条件,这为预防脑动脉瘤的破裂提供了理论依据..     

污染环境下单种群模型生存阈值

本论文研究了污染环境下毒素对单种群生存的影响。在环境容纳量较小的假设下建立了生物种群模型,在该模型中不但考虑了环境毒素浓度对生物个体生存的影响,还考虑了生物个体从食物链中吸收的毒素对其影响。通过研究得到种群一致持续生存和若平均持续生存的充分条件,同时得到种群持续生存依赖于模型参数和生物个体体内毒素净化率的某些充分条件．     

Optimal risk management of human alveolar echinococcosis with vermifuge

In this study, we develop a bioeconomic model of human alveolar echinococcosis (HAE) and formulate the optimal strategies for managing the infection risks in humans by applying optimal control theory. The model has the following novel features: (i) the complex transmission cycle of HAE has been tractably incorporated into the framework of optimal control problems and (ii) the volume of vermifuge spreading to manage the risk is considered a control variable. With this model, we first obtain the stability conditions for the transmission dynamics under the condition of constant control. Second, we explicitly introduce a control variable of vermifuge spreading into the analysis by considering the associated control costs. In this optimal control problem, we have successfully derived a set of conditions for a bang-bang control and singular control, which are mainly characterized by the prevalence of infection in voles and foxes and the remaining time of control. The analytical results are demonstrated by numerical analysis and we discuss the effects of the parameter values on the optimal strategy and the transmission cycle. We find that when the prevalence of infection in foxes is low and the prevalence of infection in voles is sufficiently high, the optimal strategy is to expend no effort in vermifuge spreading.     

Multi-compartmental modelling and experimental design for glucose transport studies in insulin-resistant rats

Many pathologies are associated with abnormalities of glucose metabolism or with perturbations of its transport (type 2 diabetes or insulin-resistance). The pre-diabetic state is characterised by a state of insulin-resistance, in others words a defect of glucose transport in insulin-sensible tissues, such as muscles and adipose tissues. The mathematical modelling of experimental data can be an excellent method to explore the mechanisms implied in the studied biological phenomenon. Thus, starting from a symbolic formulation like the compartmental modelling, it can be possible to develop a theoretical basis for the observation and to consider the best-adapted experiments for the study. We showed with mathematical models that [123I]-6-deoxy-6-iodo-D-glucose (6-DIG), shown as a tracer of glucose transport in vitro, could point out this transport abnormality. To quantify the insulin resistance, we estimated the fractional transfer coefficients of 6-DIG from the blood to the organs. We realised many studies to lead to a satisfying model; special attention has been paid to the precision of the parameter to select the best model. The results showed that by associating experimental data obtained with 6-DIG activities and an adapted mathematical model, discriminating parameters (in and out fractional transfer coefficients) between the two groups (control and insulin-resistant rats) could be pointed out.     

Design and analysis of a solar reactor for anaerobic wastewater treatment

The aim of this research was to design a solar heated reactor system to enhance the anaerobic treatment of wastewater or biological sludge at temperatures higher than the ambient air temperature. For the proposed reactor system, the solar energy absorbed by flat plate collectors was transferred to a heat storage tank, which continuously supplied an anaerobic-filter reactor with water at a maximum temperature of 35 degrees C. The packed reactor was a metallic cylindrical tank with a peripheral twin-wall enclosure. Inside this enclosure was circulated warm water from the heat storage tank. Furthermore, a mathematical model was developed for the prediction of the temperature distribution within the reactor under steady state conditions. Preliminary results based on model simulations performed with meteorological data from various geographical regions of the world suggested that the proposed solar reactor system could be a promising and environmentally friendly approach for anaerobic treatment of wastewater and biological sludge.     

Contribution of a mathematical modelling approach to the understanding of the ovarian function

The biological meaning of folliculogenesis is to free fertilisable oocytes at the time of ovulation. We approached the study of the control of follicular development at the level of follicular granulosa cells, on the experimental as well as mathematical modelling grounds. We built a mathematical model allowing for the processes of proliferation, differentiation and apoptosis. State variables correspond to the numbers of cells undergoing these different processes, while control variables correspond to the cellular transition rates. The model results raised the notion of proliferative resources, which leads to consider the optimal management of these resources and has motivated the settling of an experiment investigating the changes in the growth fraction within the granulosa throughout terminal development. We are now investigating the way gonadotrophins, and especially FSH, operate on granulosa cells, in order to account for the hormonal control of the divergent commitment of granulosa cells towards either proliferation, differentiation or apoptosis. We are thus focusing on the dynamics of cAMP production, which appears to be a keypoint in FSH signal transduction.     

Optimal control for selected cancer chemotherapy ODE models: a view on the potential of optimal schedules and choice of objective function

In this article, four different mathematical models of chemotherapy from the literature are investigated with respect to optimal control of drug treatment schedules. The various models are based on two different sets of ordinary differential equations and contain either chemotherapy, immunotherapy, anti-angiogenic therapy or combinations of these. Optimal control problem formulations based on these models are proposed, discussed and compared. For different parameter sets, scenarios, and objective functions optimal control problems are solved numerically with Bock’s direct multiple shooting method.In particular, we show that an optimally controlled therapy can be the reason for the difference between a growing and a totally vanishing tumor in comparison to standard treatment schemes and untreated or wrongly treated tumors. Furthermore, we compare different objective functions. Eventually, we propose an optimization-driven indicator for the potential gain of optimal controls. Based on this indicator, we show that there is a high potential for optimization of chemotherapy schedules, although the currently available models are not yet appropriate for transferring the optimal therapies into medical practice due to patient-, cancer-, and therapy-specific components.     

Biodegradability of tannin-containing wastewater from leather industry

Tannins occur commonly in the wastewaters from forestry, plant medicine, paper and leather industries. The treatment of this kind of wastewaters, including settling and biodegradation, is usually difficult because tannins are highly soluble in water and would inhibit the growth of microorganisms in activated sludge. The objective of this study is to investigate biodegradability of tannin-containing wastewaters, so as to characterize the pollution properties of such wastewaters and provide a reference for their biological treatment in wastewater treatment plants. The research was typified by using the wastewater collected from vegetable tanning process in leather industry. A model was developed to describe the activated sludge process, and the biodegradation kinetics of vegetable tanning wastewater (VET wastewater) was studied. It was found that the biodegradability of tannin-containing wastewater varies heavily with the content of tannins in wastewater. The biodegradation of VET wastewater with tannin content around 4,900 mg/l occurred inefficiently due to the inhibition of tannins to the activated sludge process, and only 34.7% of biodegradation extent was reached in 14 days of incubation. The optimal biodegradability of VET wastewater was observed when its tannin content was diluted to 490 mg/l, where the COD and tannin removals reached 51.3% and 45.1% respectively in 6 days. Hence, it is suggested that a proper control of tannin content is necessary to achieve an effective biodegradation of tannin-containing wastewaters in wastewater treatment plants.     

Optimal Control in a Model of Dendritic Cell Transfection Cancer Immunotherapy

We construct a population dynamics model of the competition among immune system cells and generic tumor cells. Then, we apply the theory of optimal control to find the optimal schedule of injection of autologous dendritic cells used as immunotherapeutic agent.The optimization method works for a general ODE system and can be applied to find the optimal schedule in a variety of medical treatments that have been described by a mathematical model.     

Microflora from leaf debris is suitable for treatment of starch industry wastewater

Biological treatment of industrial waste is a widely practiced technique that generates comparatively less environmentally hazardous waste than other chemical treatment processes. Wet milling of maize generates huge amount of wastewater (5 m³/ton) of low pH with organic matter and nutrients. Anaerobic methanogenic and aerobic bacteria are mostly highly sensitive to low pH. The treatment of wastewater causes huge cost of chemical neutralization or hydraulic recirculation for maintaining neutral pH. In the present study, different microbial consortia isolated from cow dung, active sludge from an anaerobic reactor for treatment of industrial wastewater, and leaf debris from benthic soil were screened for tolerance against low pH and for potential of chemical oxygen demand (COD) removal in order to find out an alternative microbial population for industrial water treatment at low pH. The most effective consortia found from leaf debris were further investigated for optimal operation. The microscopic analysis of leaf debris sludge showed abundance of Gram‐negative methanococci, which was found tolerant to low pH in plate culture method. On further investigation for COD removal from starch industry effluent, they were found to be most effective at pH 5 with highest COD removal rate of 70% and lowest biomass generation of 81%. Hence, it was concluded that the low pH‐tolerant methanogen bacteria, enriched from leaf debris sludge, is highly beneficial for anaerobic treatment of wastewater from several industries including corn starch industry by reducing cost of operation for neutralization to neutral pH and through reducing excess waste sludge production by the treatment system.     

Strategies and Practices for Sustainable Use of Water in Industrial Papermaking Processes

The problem of water use in industrial sectors is analyzed in the context of sustainability, and the manufacturing process as one of the most important water consumers is scrutinized. Some practices of water conservation and reduction of water consumption, as well as the implementation of advanced treatment processes for wastewater recycling are considered under the general aspect of process integration and control in order to minimize environmental discharges. Reduction of effluents in papermaking processes are discussed as well as preventative approaches. It is emphasized that four groups of methods can be applied in order to minimize water consumption: process changes, water reuse, regeneration reuse and regeneration recycling. Minimizing the water consumption and thus reducing the hydraulic load may help to improve effluent treatment, reduce operating costs and, when subsequent disposal is to sewers, minimize effluent disposal charges. The data presented show that initially producing less effluents will reduce the demands on the effluent treatment plant and thus save both money and efforts. Separation techniques for minimizing effluents and effluent loads are presented, as well as potential practices to ensure a sustainable use of water, grouped as short term and long term measures. Prevention measures for water consumption in a Romanian paper mill are considered and discussed in relation with best available techniques (BAT), taking into account both economical and environmental benefits.     

年龄相关的种群系统的最优生育率控制

讨论了一类与年龄相关的时变种群系统最优生育率控制的非线性问题,利用Ba- nach空间的Saks-Masur引理,证明了系统最优生育率控制的存在唯一性,并得到了控制为最优的必要条件和最优性组．所得结果可为种群系统控制问题的实际研究提供必要的数学理论基础．