Optimal control of epidemics with limited resources

We extend the existing work on the time-optimal control of the basic SIR epidemic model with mass action contact rate. Previous results have focused on minimizing an objective function that is a linear combination of the cost associated with using control and either the outbreak size or the infectious burden. We instead, provide analytic solutions for the control that minimizes the outbreak size (or infectious burden) under the assumption that there are limited control resources. We provide optimal control policies for an isolation only model, a vaccination only model and a combined isolation–vaccination model (or mixed model). The optimal policies described here contain many interesting features especially when compared to previous analyses. For example, under certain circumstances the optimal isolation only policy is not unique. Furthermore the optimal mixed policy is not simply a combination of the optimal isolation only policy and the optimal vaccination only policy. The results presented here also highlight a number of areas that warrant further study and emphasize that time-optimal control of the basic SIR model is still not fully understood.     

广义Logistic模型的捕获优化问题

以王寿松所提出的广义Ｌｏｇｉｓｔｉｃ模型为基础,讨论单种群生物资源的捕获优化问题,分析了被开发生物种群的动力学性质。在单位捕获努力量假定下,以最大可持续捕获量为管理目标,确定了线性捕获下的最优捕获策略,得到了最优捕获努力量,最大可持续收获及相应的最优种群水平的显式表达式,包括著名的Ｓｃｈａｅｆｅｒ模型作为特例,推广了相应的结果。     

甘南高寒草甸退化对植物功能群多样性与物种多度分布的影响

为分析甘南高寒草甸植物功能群多样性及其物种多度分布对退化的响应,探讨其多样性的形成与维持机制,采用样地调查法收集数据,并使用物种多度模型对其进行拟合分析。结果表明:随着退化程度的加深,植被优势种以禾草科和莎草科植物为主,逐渐变为杂草功能群植物为主,杂草功能群在群落多样性分布中起着主导作用;全部物种多度分布随着退化程度的加深发生变化,其中,无退化草甸的最优拟合模型是VOLKOV,轻度退化草甸的最优拟合模型是GEO,中度退化草甸和重度退化草甸的最优拟合模型为BRO,资源分配模式由随机分配转向固定分配的分配模式;禾草功能群的最优拟合模型以生态位模型为主,资源分配方式由固定分配和随机分配共同主导,豆科功能群的最优拟合模型是BRO,资源分配方式以固定分配为主,杂草功能群的最优拟合模型从中性模型向生态位模型转变,与全部物种多度分布的最优模型基本一致,且资源分配由随机分配向固定分配转变,可以认为杂草功能群是影响群落物种多度分布的主要原因,但是禾草和豆科功能群的贡献也不可忽视。     

Optimal sampling and estimation strategies under the linear model

In some cases model-based and model-assisted inferences canlead to very different estimators. These two paradigms are notso different if we search for an optimal strategy rather thanjust an optimal estimator, a strategy being a pair composedof a sampling design and an estimator. We show that, under alinear model, the optimal model-assisted strategy consists ofa balanced sampling design with inclusion probabilities thatare proportional to the standard deviations of the errors ofthe model and the Horvitz–Thompson estimator. If the heteroscedasticityof the model is 'fully explainable’ by the auxiliary variables,then this strategy is also optimal in a model-based sense. Moreover,under balanced sampling and with inclusion probabilities thatare proportional to the standard deviation of the model, thebest linear unbiased estimator and the Horvitz–Thompsonestimator are equal. Finally, it is possible to construct asingle estimator for both the design and model variance. Theinference can thus be valid under the sampling design and underthe model.     

Assessing the Performance of Neural Encoding Models in the Presence of Noise

An analytical method is introduced for evaluating the performance of neural encoding models. The method addresses a critical question that arises during the course of the development and validation of encoding models: is a given model near optimal in terms of its accuracy in predicting the stimulus-elicited responses of a neural system, or can the predictive accuracy be improved significantly by further model development? The evaluation method is based on a derivation of the minimum mean-square error between actual responses and modeled responses. It is formulated as a comparison between the mean-square error of the candidate model and the theoretical minimum mean-square error attainable through an optimal model for the system. However, no a priori information about the nature of the optimal model is required. The theoretically minimum error is determined solely from the coherence function between pairs of system responses to repeated presentations of the same dynamic stimulus. Thus, the performance of the candidate model is judged against the performance of an optimal model rather than against that of an arbitrarily assumed model. Using this method, we evaluated a linear model for neural encoding by mechanosensory cells in the cricket cercal system. At low stimulus intensities, the best-fit linear model of encoding by single cells was found to be nearly optimal, even though the coherence between stimulus-response pairs (a commonly used measure of system linearity) was low. In this low-stimulus-intensity regime, the mean square error of the linear model was on the order of the power of the cell responses. In contrast, at higher stimulus intensities the linear model was not an accurate representation of neural encoding, even though the stimulus-response coherence was substantially higher than in the low-intensity regime.     

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.     

微生物连续培养模型的最优溢流量控制

本文使用最优控制理论研究了微生物连续培养模型的最优溢流量控制问题,导出了一种关于微生物浓度的线性反馈控制律．在该控制律下,可保证微生物连续培养模型能跟踪理想微生物浓度ｘｄ．     

Robust and efficient design of experiments for the Monod model

In this paper the problem of designing experiments for the Monod model, which is frequently used in microbiology, is studied. The model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics, environmental research, pharmacokinetics, and plant physiology. The designs presented so far in the literature are local optimal designs, which depend sensitively on a preliminary guess of the unknown parameters, and are for this reason in many cases not robust with respect to their misspecification. Uniform designs and maximin optimal designs are considered as a strategy to obtain robust and efficient designs for parameter estimation. In particular, standardized maximin D- and E-optimal designs are determined and compared with uniform designs, which are usually applied in these microbiological models. It is demonstrated that maximin optimal designs are substantially more efficient than uniform designs. Parameter variances can be decreased by a factor of two by simply sampling at optimal times during the experiment. Moreover, the maximin optimal designs usually provide the possibility for the experimenter to check the model assumptions, because they have more support points than parameters in the Monod model.     

Optimal Vaccination Policies for an SIR Model with Limited Resources

The purpose of the paper is to use analytical method and optimization tool to suggest a vaccination program intensity for a basic SIR epidemic model with limited resources for vaccination. We show that there are two different scenarios for optimal vaccination strategies, and obtain analytical solutions for the optimal control problem that minimizes the total cost of disease under the assumption of daily vaccine supply being limited. These solutions and their corresponding optimal control policies are derived explicitly in terms of initial conditions, model parameters and resources for vaccination. With sufficient resources, the optimal control strategy is the normal Bang–Bang control. However, with limited resources, the optimal control strategy requires to switch to time-variant vaccination.     

Reducing model complexity of the general Markov model of evolution

The selection of an optimal model for data analysis is an important component of model-based molecular phylogenetic studies. Owing to the large number of Markov models that can be used for data analysis, model selection is a combinatorial problem that cannot be solved by performing an exhaustive search of all possible models. Currently, model selection is based on a small subset of the available Markov models, namely those that assume the evolutionary process to be globally stationary, reversible, and homogeneous. This forces the optimal model to be time reversible even though the actual data may not satisfy these assumptions. This problem can be alleviated by including more complex models during the model selection. We present a novel heuristic that evaluates a small fraction of these complex models and identifies the optimal model.     

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.     

Optimization of harvesting age in an integral age-dependent model of population dynamics

The paper deals with optimal control in a linear integral age-dependent model of population dynamics. A problem for maximizing the harvesting return on a finite time horizon is formulated and analyzed. The optimal controls are the harvesting age and the rate of population removal by harvesting. The gradient and necessary condition for an extremum are derived. A qualitative analysis of the problem is provided. The model shows the presence of a zero-investment period. A preliminary asymptotic analysis indicates possible turnpike properties of the optimal harvesting age. Biological interpretation of all results is provided.     

Application of radiography of computed tomography in non-small cell lung cancer using prognosis model

ObjectiveStudying the diagnostic value of CT imaging in non-small cell lung cancer (NSCLC), and establishing a prognosis model combined with clinical characteristics is the objective, so as to provide a reference for the survival prediction of NSCLC patients.MethodCT scan data of NSCLC 200 patients were taken as the research object. Through image segmentation, the radiology features of CT images were extracted. The reliability and performance of the prognosis model based on the optimal feature number of specific algorithm and the prognosis model based on the global optimal feature number were compared.Results30-RELF-NB (30 optimal features, RELF feature selection algorithm and NB classifier) has the highest accuracy and AUC (area under the subject characteristic curve) in the prognosis model based on the optimal features of specific algorithm. Among the prognosis models based on global optimal features, 25-NB (25 global optimal features, naive Bayes classification algorithm classifier) has the highest accuracy and AUC. Compared with the prediction model based on feature training of specific feature selection algorithm, the overall performance and stability of the prediction model based on global optimal feature are higher.ConclusionThe prognosis model based on the global optimal feature established in this paper has good reliability and performance, and can be applied to the CT radiology of NSCLC.     

Effort fluctuations in a harvest model with random prices

The effect of a single increase in the unit price of biomass on the optimal harvest policy for an exploited population is studied numerically. The price of a unit is assumed constant until a random time, when the price increases by a given amount. The optimal policy corresponding to the expected return is computed from the Bellman equation of dynamic programming. The results are compared with a model in which prices remain constant as well as a “well-timed” model in which the price increases at the expected increase time of the random case. Both optimal expected return and optimal policy computed from the deterministic models may differ substantially from that calculated from the random model, particularly if marginal costs are large. The emphasis is on numerical computation.     

Optimal coordination and control of posture and locomotion.

This paper presents a theoretical model of stability and coordination of posture and locomotion, together with algorithms for continuous-time quadratic optimization of motion control. Explicit solutions to the Hamilton-Jacobi equation for optimal control of rigid-body motion are obtained by solving an algebraic matrix equation. The stability is investigated with Lyapunov function theory, and it is shown that global asymptotic stability holds. It is also shown how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. The solution describes motion strategies of minimum effort and variance. The proposed optimal control is formulated to be suitable as a posture and stance model for experimental validation and verification. The combination of adaptive and optimal control makes this algorithm a candidate for coordination and control of functional neuromuscular stimulation as well as of prostheses.     

A mathematical model of cancer chemotherapy with an optimal selection of parameters

An optimal parameter selection model of cancer chemotherapy is presented which describes the treatment of a tumor over a fixed period of time by the repeated administration of a single drug. The drug is delivered at evenly spaced intervals over the treatment period at doses to be selected by the model. The model constructs a regimen that both minimizes the tumor population at the end of the treatment and satisfies constraints on the drug toxicity and intermediate tumor size. Numerical solutions show that an optimal regimen withholds the bulk of the doses until the end of the treatment period. When a drug used is of either moderate or low effectiveness, an optimal regimen is superior to a schedule that delivers all of the drug at the beginning of the treatment. This study questions whether the current method for the administration of chemotherapy is optimal and suggests that alternative regimens should be considered.     

A patient risk model to determine the optimal output constancy check frequency for a radiotherapy machine

PurposeThe output constancy check, a basic quality control (QC) item for radiotherapy machines, is performed daily according to suggestions in technical reports by experienced experts. In this study, a patient risk model was built to determine the optimal frequency of an output constancy check for a specific radiotherapy machine.Methods and materialsThe method was based on the patient risk model and comprised three steps: 1) the power function graph was used to select a proper QC rule and the average number of QC measurements per QC rule evaluation. 2) The optimal QC frequency was determined by the minimum integer value of expected patients treated between QC measurements. 3) The individual control chart (I-Chart) was used to evaluate the effectiveness of the model. The model was implemented on the output constancy check of a Tomotherapy machine.ResultsThe QC rule with the limits set to the mean ± 3 standard deviations and 5 measurements per QC were selected according to the power function graph. The optimal frequency was observed every 21 patients. The I-Chart showed that the optimal frequency detected the machine failure earlier compared to the conventional daily frequency. The model could monitor whether Tomotherapy machine was in good condition and predicted the time to adjust the machine.ConclusionsThe optimal output constancy check frequency of a radiotherapy machine is determined by the number of patients, which uses patient risk model. The optimal frequency is superior to the conventional daily frequency in identifying machine failure earlier.     

Modeling and optimal control of insect pest populations

A distributed-parameter population dynamics model is developed specifically for use with variational optimization techniques. The objective is to develop a modeling/ optimization technique that will permit the development of optimal control policies which minimize combined costs of pest control and economic-yield loss. The model results and optimal control policies are continuous and distributed in time and in insect age. The technique is applied to a study of control by pesticide of the southern green stink bug, Nezara viridula (Linnaeus), a major pest of soybeans in the South. In this case study, the model results agree well with field-survey results, while the optimal control trajectories are reasonable and suggest several avenues for further study.     

React or wait: which optimal culling strategy to control infectious diseases in wildlife

We applied optimal control theory to an SI epidemic model to identify optimal culling strategies for diseases management in wildlife. We focused on different forms of the objective function, including linear control, quadratic control, and control with limited amount of resources. Moreover, we identified optimal solutions under different assumptions on disease-free host dynamics, namely: self-regulating logistic growth, Malthusian growth, and the case of negligible demography. We showed that the correct characterization of the disease-free host growth is crucial for defining optimal disease control strategies. By analytical investigations of the model with negligible demography, we demonstrated that the optimal strategy for the linear control can be either to cull at the maximum rate at the very beginning of the epidemic (reactive culling) when the culling cost is low, or never to cull, when culling cost is high. On the other hand, in the cases of quadratic control or limited resources, we demonstrated that the optimal strategy is always reactive. Numerical analyses for hosts with logistic growth showed that, in the case of linear control, the optimal strategy is always reactive when culling cost is low. In contrast, if the culling cost is high, the optimal strategy is to delay control, i.e. not to cull at the onset of the epidemic. Finally, we showed that for diseases with the same basic reproduction number delayed control can be optimal for acute infections, i.e. characterized by high disease-induced mortality and fast dynamics, while reactive control can be optimal for chronic ones.     

Optimal foraging area: Size and allocation of search effort

A model is derived for the optimal spatial allocation of foraging effort for an animal returning with food to a central place in a uniform habitat. The forager is assumed to maximize its yield of food during a given period. Foraging effort is expended on search for food, and on transportation to the central place. It is shown that the allocation of search has been optimal if and only if the “marginal cost” of additional food is equal throughout the foraging area when the period has elapsed. The model is used to predict the optimal area radius and allocation of search time. With realistic parameter values, the optimal time per unit area roughly decreases linearly with the distance from the central place. The influence of food density and forager characteristics is examined.