Asymptotic existence of proportionally fair allocations

Fair division has long been an important problem in the economics literature. In this note, we consider the existence of proportionally fair allocations of indivisible goods, i.e., allocations of indivisible goods in which every agent gets at least her proportionally fair share according to her own utility function. We show that when utilities are additive and utilities for individual goods are drawn independently at random from a distribution, proportionally fair allocations exist with high probability if the number of goods is a multiple of the number of agents or if the number of goods grows asymptotically faster than the number of agents.     

Modelling of tumor cells regression in response to chemotherapeutic treatment

The proposed model describes the interaction among normal, immune and tumor cells in a tumor with a chemotherapeutic drug, using a system of four coupled partial differential equations. The dimensions of the tumor and initial conditions of tumor cells are chosen under the assumption that the tumor is already large enough in size to be detectable with the available clinical devices. The pattern of distribution of tumor cells is drafted on the basis of clinical observations. The stability of the system is established with tumor and tumor-free equilibria. The process of tumor regression with the introduction of different diffusion coefficients of tumor and immune cells is considered along with normal cells of tissue without any diffusive movement. It is shown that the results of chemotherapy treatment are in agreement with Jeff’s phenomenon. The response of three different levels of immune system strength to the pulsed chemotherapy are investigated. It is observed that the tumor performs better if a chemotherapeutic drug is injected near the invasive fronts of the tumor.     

A mixed radiotherapy and chemotherapy model for treatment of cancer with metastasis

In this paper, a mathematical model of cancer treatment, in the form of a system of ordinary differential equations, by chemotherapy and radiotherapy where there is metastasis from a primary to a secondary site has been proposed and analyzed. The interaction between immune cells and cancer cells has been examined, and the chemotherapy agent has been considered as a predator on both normal and cancer cells. The metastasis may be time delayed. For better investigation of the treatment process and based on physical investigation, the immanent effects of inputs on cancer dynamic have been investigated. It is supposed that the interaction between NK cells and tumor cells changes during the chemotherapy. This novel approach is useful not only to gain a broad understanding of the specific system dynamics but also to guide the development of combination therapies. The analysis is carried out both analytically (where possible) and numerically. By considering such immanent effects, the tumor‐free equilibrium point will be stable at the end of treatment, and the tumor can not recur again, and the patient will totally recover. So, the present analysis suggests that a proper treatment method should change the dynamics of the cancer instead of only reducing the population of cancer cells. Copyright © 2016 John Wiley & Sons, Ltd.     

Interval-valued probability density estimation based on quasi-continuous histograms: Proof of the conjecture

The sensitivity of histogram computation to the choice of a reference interval and number of bins can be attenuated by replacing the crisp partition on which the histogram is built by a fuzzy partition. This involves replacing the crisp counting process by a distributed (weighted) voting process. The counterpart to this low sensitivity is some confusion in the count values: a value of 10 in the accumulator associated with a bin can mean 10 observations in the bin or 40 observations near the bin. This confusion can bias the statistical decision process based on such a histogram. In a recent paper, we proposed a method that links the probability measure associated with any subset of the reference interval with the accumulator values of a fuzzy partition-based histogram. The method consists of transferring counts associated with each bin proportionally to its interaction with the considered subset. Two methods have been proposed which are called precise and imprecise pignistic transfer. Imprecise pignistic transfer accounts for the interactivity of two consecutive cells in order to propagate, in the estimated probability measure, counting confusion due to fuzzy granulation. Imprecise pignistic transfer has been conjectured to include precise pignistic transfer. The present article proposes a proof of this conjecture.     

A survey of optimization models on cancer chemotherapy treatment planning

While chemotherapy is an effective method for treating cancers such as colorectal cancer, its effectiveness may be dampened by the drug resistance and it may have significant side effects due to the destruction of normal cells during the treatment. As a result, there is a need for research on choosing an optimal chemotherapy treatment plan that minimizes the number of cancerous cells while ensuring that the total toxicity is below an allowable limit. In this paper, we summarize the mathematical models applied to the optimal design of the cancer chemotherapy. We first elaborate on a typical optimization model and classify relevant literature with respect to modeling methods: Optimal control model (OCM) and others. We further classify the OCM models with respect to the solution method used. We discuss the limitations of the existing research and provide several directions for further research in optimizing chemotherapy treatment planning.     

Effect of mass transfer and diffusion of nanofluid on the thermal ablation of malignant cells during magnetic hyperthermia

Malignant cells can be ablated by a specific treatment temperature during magnetic hyperthermia, which is induced by the power dissipation of magnetic nanoparticles (MNPs) inside tumor region under an alternating magnetic field. MNPs contained in nanofluid need to be transferred to tumor region before therapy can begin, and one of the most prominent methods is direct injection. Although different aspects of this area are covered in literature, the study of models which consider the combined effects of nanofluid transport and heat generation on the ablation of malignant cells still lacks enough attention. A complete computational model is developed in this paper to evaluate the survival rate of malignant cells for a proposed geometric model when intratumoral injection of MNPs is considered. The mathematical model incorporates the transport of nanofluid inside the bio-tissue, the heat generation of MNPs during ablation, the heat transfer of bio-tissue, and the cell death probability based on the Arrhenius model. The concentration distribution of nanofluid and the treatment temperature profile inside bio-tissue are obtained by considering the finite element method with the proposed boundary and initial conditions. Simulation results demonstrate that the death rate of malignant cells can be considerably improved when a proper critical power dissipation of MNPs is designed and enough diffusion duration is considered for therapy. With further developments, the model may be used for the planning of magnetic hyperthermia.     

Modeling treatment of cancer using virotherapy with generalized logistic growth of tumor cells

Abstract

Virotherapy is an effective strategy in cancer treatment. It eliminates tumor cells without harming the healthy cells. In this article, a deterministic mathematical model to understand the dynamics of tumor cells in response to virotherapy is formulated and analyzed by incorporating cytotoxic T lymphocytes (CTLs). The basic reproduction number and the immune response reproduction number are computed and different equilibria of the proposed model are found. The local stability of different equilibria is discussed in detail. Further, the proposed model is extended to stochastic model. Numerical simulation is performed for both deterministic and stochastic models. It is observed that when both the reproduction numbers are greater than one, which corresponds to existence of unique nontrivial equilibrium point, dynamics of deterministic and stochastic models are almost same. The deterministic model shows a very complex dynamics when one or both the reproduction numbers are below one. The system exhibits both backward bifurcation and Hopf-bifurcation for suitable sets of parameters and in this situation it is not easy to predict the dynamics of cancer cells and virus particles. The existence of backward bifurcation demonstrates the fact that partial success of virotherapy can be achieved even if the immune response reproduction number is less than one.     

The maximum proportionally modular numerical semigroup with given multiplicity and ratio

We prove that the set of all proportionally modular numerical semigroups with fixed multiplicity and ratio has a maximum (with respect to set inclusion). We show that this maximum is a maximal embedding dimension numerical semigroup, for which we explicitly calculate its minimal system of generators, Frobenius number and genus.     

A cellular automata model of chemotherapy effects on tumour growth: targeting cancer and immune cells

The effects of therapy on avascular cancer development based on a stochastic cellular automata model are considered. Making the model more compatible with the biology of cancer, the following features are implemented: intrinsic resistance of cancerous cells along with drug-induced resistance, drug-sensitive cells, immune system. Results are reported for no treatment, discontinued treatment after only one cycle of chemotherapy, and periodic drug administration therapy modes. Growth fraction, necrotic fraction, and tumour volume are used as output parameters beside a 2-D graphical growth presentation. Periodic drug administration is more effective to inhibit the growth of tumours. The model has been validated by the verification of the simulation results using in vivo literature data. Considering immune cells makes the model more compatible with the biological realities. Beside targeting cancer cells, the model can also simulate the activation of the immune system to fight against cancer.

Abbreviations CA: cellular automata; DSC: drug sensitive cell; DRC: drug resistant cell; GF: growth fraction; NF: necrotic fraction; ODE: ordinary differential equation; PDE: partial differential equation; SCAM: The proposed stochastic cellular automata model     

Dynamical analysis of chemotherapy models with time-dependent infusion

Classical mathematical models for chemotherapy assume a constant infusion rate of the chemotherapy agent. However in reality the infusion rate usually varies with respect to time, due to the natural (temporal or random) fluctuation of environments or clinical needs. In this work we study a non-autonomous chemotherapy model where the injection rate and injection concentration of the chemotherapy agent are time-dependent. In particular, we prove that the non-autonomous dynamical system generated by solutions to the non-autonomous chemotherapy system possesses a pullback attractor. In addition, we investigate the detailed interior structures of the pullback attractor to provide crucial information on the effectiveness of the treatment. The main analytical tool used is the theory of non-autonomous dynamical systems. Numerical experiments are carried out to supplement the analysis and illustrate the effectiveness of different types of infusions.     

Endogenous optimization fisheries models

This paper deals with a special class of fisheries models referred to as endogenous optimization models. The distinctive feature of these models is that behaviour of the agents in the model is not predetermined by exogenous behavioural rules. In endogenous optimization models, the model agents are merely furnished with objectives such as profit or utility maximization. Given these objectives and the various constraints determined by the state of the model at each point of time, the agents solve their maximization problem. The corresponding values of their control variables then constitute their behaviour.Having generated individual agents' behaviour by endogenous optimization, summing over agents yields aggregate behaviour. Aggregate behaviour must conform with the overall constraints of the model, be they physical or otherwise. Within the market system, individual behaviour or rather plans are made compatible via changes in relative prices. Therefore, outside equilibrium, behavioural plans must be repeatedly modified to become mutually compatible. This implies iteratively solving the maximization problem of a number of different agents. Endogenous optimization models therefore tend to be computationally very demanding.Clearly, the basic principles of endogenous optimization are just as applicable to any model involving maximizing agents.     

Stochastic interactions increase cooperation in a spatial Prisoner's Dilemma

This article implements the spatial Prisoner's Dilemma (PD) as an agent‐based model. Many previous models have assumed that agents in a spatial PD interaction exclusively and deterministically within their von Neumann neighborhood. The model presented here introduces stochastic interactions within a subset of the von Neumann neighborhood. This implementation allows a direct comparison of the effect of stochastic interactions relative to deterministic interactions on the level of cooperation that emerges in the system. The results show that when holding the total number of interactions agents participate in each round constant, allowing agents to interact stochastically increases cooperation in the system relative to deterministic interactions. © 2014 Wiley Periodicals, Inc. Complexity 21: 82–92, 2015     

Numerical optimisation of chemotherapy dosage under antiangiogenic treatment in the presence of drug resistance

We consider a two-compartment model of chemotherapy resistant tumour growth under angiogenic signalling. Our model is based on the one proposed by Hahnfeldt et al. (1999), but we divide tumour cells into sensitive and resistant subpopulations. We study the influence of antiangiogenic treatment in combination with chemotherapy. The main goal is to investigate how sensitive are the theoretically optimal protocols to changes in parameters quantifying the interactions between tumour cells in the sensitive and resistant compartments, that is, the competition coefficients and mutation rates, and whether inclusion of an antiangiogenic treatment affects these results. Global existence and positivity of solutions and bifurcations (including bistability and hysteresis) with respect to the chemotherapy dose are studied. We assume that the antiangiogenic agents are supplied indefinitely and at a constant rate. Two optimisation problems are then considered. In the first problem a constant, indefinite chemotherapy dose is optimised to maximise the time needed for the tumour to reach a critical (fatal) volume. It is shown that maximum survival time is generally obtained for intermediate drug dose. Moreover, the competition coefficients have a more visible influence on survival time than the mutation rates. In the second problem, an optimal dosage over a short, 30-day time period, is found. A novel, explicit running penalty for drug resistance is included in the objective functional. It is concluded that, after an initial full-dose interval, an administration of intermediate dose is optimal over a broad range of parameters. Moreover, mutation rates play an important role in deciding which short-term protocol is optimal. These results are independent of whether antiangiogenic treatment is applied or not.     

On strategies on a mathematical model for leukemia therapy

A mathematical model for leukemia therapy based on the Gompertzian law of cell growth is studied. It is assumed that the chemotherapeutic agents kill leukemic as well as normal cells.Effectiveness of the medicine is described in terms of a therapy function. Two types of therapy functions are considered: monotonic and non-monotonic. In the former case the level of the effect of the chemotherapy directly depends on the quantity of the chemotherapeutic agent. In the latter case the therapy function achieves its peak at a threshold value and then the effect of the therapy decreases. At any given moment the amount of the applied chemotherapeutic is regulated by a control function with a bounded maximum. Additionally, the total quantity of chemotherapeutic agent which can be used during the treatment process is bounded too.The problem is to find an optimal strategy of treatment to minimize the number of leukemic cells while at the same time retaining as many normal cells as possible.With the help of Pontryagin’s Maximum Principle it was proved that the optimal control function has at most one switch point in both monotonic and non-monotonic cases for most relevant parameter values.A control strategy called alternative is suggested. This strategy involves increasing the amount of the chemotherapeutical medicine up to a certain value within the shortest possible period of time, and holding this level until the end of the treatment.The comparison of the results from the numerical calculation using the Pontryagin’s Maximum Principle with the alternative control strategy shows that the difference between the values of cost functions is negligibly small.     

On sequential and simultaneous contributions under incomplete information

When contributors to a common cause (or public good) are uncertain about each others’ valuations, early contributors are likely to be cautious in free-riding on future contributors. Contrary to the case of complete information, when contributors have independent private valuations for the public good, the expected total contribution generated in a sequential move game may be higher than in a simultaneous move game. This is established in a conventional framework with quasi-linear utility where agents care only about the total provision of the public good (rather than individual contribution levels) and there is no non-convexity in the provision of the public good. We allow for arbitrary number of agents and fairly general distribution of types.     

Cayley-like representations are for all algebras, not merely groups

Cayley 's Theorem represents an arbitrary group as a set of permutations with the group operation captured by the composition of permutations. A few other examples with related representations are monoids, Boolean algebras and Menger algebras, permutations now being replaced by functions with one or more arguments. Although Cayley-like representations appear to be rare, this article shows that they are not. The idea is to represent the elements of an arbitrary algebra by multivariable functions, and its operations by particular compositions of these functions. Any finite algebra can be so represented,and so can any variety generated by one finite subdirectly irreducible algebra. It will follow that these varieties are Cayley-like: semilattices, distributive lattices, median algebras, elementary Abelian p -groups (for fixed p), and those generated by a primal algebra. If the definition of Cayley-like is stretched to allow the representing functions to have an infinite number of arguments, then all algebras are Cayley-like.     

Route planning for agent-based information retrieval

This article focuses on Mobile Agents and their use for information retrieval. A multi-agent system is considered; a number of agents are involved in a collective effort to retrieve distributed data from network nodes. Increasing the number of agents may speed-up information retrieval but is burdensome to performance. Initiating with a given number of agents, our objective is to determine the routes of the agents so that the task completion time is minimized. Two known and one new polynomial-time algorithms are tested that produce near-optimal solutions. Simulation results show the cases for which each one is most effective. Additionally, we study the influence of various parameters on the solution. By parametrically varying the number of agents, our method can be used to determine the minimum number that satisfies the desired trade-off between time and performance.     

逐步距离判别分析在甲状腺滤泡性腺瘤与滤泡性腺癌的判定中的运用

在认定甲状腺滤泡性腺瘤与滤泡性腺癌的细胞核形态指标存在显著差异的条件下,对两类样本通过逐步判别分析筛选出鉴别能力较强的少数几个指标,并运用距离判别分析建立了甲状腺滤泡性腺瘤与滤泡性腺癌的判别函数,为临床病理诊断提供辅助诊断方法.     

The equivalence between the Schur order and the stochastic orders generated by the scheme of allocation of particles into cells

In 1982, it was proved that the Schur partial-order relation on the set of distributions on {1, 2, ...} is equivalent to the order relation generated by the number of nonempty cell distributions in the scheme of independent allocation of particles into cells. Here it is shown that the same partialorder relation is generated by distributions of numbers of cells occupied by at least r particles for any r ≥ 2.     

The generation,propagation, and extinction of multiphases in the KdV zero‐dispersion limit

We study the multiphases in the KdV zero‐dispersion limit. These phases are governed by the Whitham equations, which are 2g + 1 quasi‐linear hyperbolic equations where g is the number of phases. We are interested in both the interaction of two single phases and the breaking of a single phase for general initial data. We analyze in detail how a double phase is generated from the interaction or breaking, how it propagates in space‐time, and how it collapses to a single phase in a finite time. The Whitham equations are known to be integrable via a hodograph transform. The crucial step in our approach is to formulate the hodograph transform in terms of the Euler‐Poisson‐Darboux solutions. Under our scheme, the zeros of the Jacobian of the transform are given by the zeros of the Euler‐Poisson‐Darboux solution. Hence, the problem of inverting the hodograph transform to give the Whitham solution reduces to that of counting the zeros of the Euler‐Poisson‐Darboux solution. © 2002 Wiley Periodicals, Inc.