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.     

An age-and-cyclin-structured cell population model for healthy and tumoral tissues

We present a nonlinear model of the dynamics of a cell population divided into proliferative and quiescent compartments. The proliferative phase represents the complete cell cycle (G ₁−S−G ₂−M) of a population committed to divide at its end. The model is structured by the time spent by a cell in the proliferative phase, and by the amount of Cyclin D/(CDK4 or 6) complexes. Cells can transit from one compartment to the other, following transition rules which differ according to the tissue state: healthy or tumoral. The asymptotic behaviour of solutions of the nonlinear model is analysed in two cases, exhibiting tissue homeostasis or tumour exponential growth. The model is simulated and its analytic predictions are confirmed numerically.     

Singular perturbation analysis of cAMP signalling in Dictyostelium discoideum aggregates

In this paper, we use singular perturbation methods to study the structure of travelling waves for some reaction-diffusion models obtained from the Martiel-Goldbeter and Goldbeter-Segel's models of cAMP signalling in Dictyostelium discoideum. As a consequence, we derive analytic formulae for quantities like wave speed, maximum concentration and other magnitudes in terms of the different biochemical constants that appear in the model.     

On the stability of separable solutions of a sexual age-structured population dynamics model

The Sharpe-Lotka-McKendrick-von Foerster equations for non-dispersing age-sex-structured populations with a harmonic mean type mating law are considered and their separable solutions are analysed. For certain forms of the demographic rates the underlying evolution equations are reduced to systems of ODEs, the long time behavior of their solutions is studied, and the stability of separable solutions is discussed. It is found that for the constant death rates and constant sex ratio of newborns with stationary birth rates this model admits one one-parameter class of separable solutions, two such classes (repeated or different) or no such ones. In the case of special forms of age-dependent birth rates, solutions of one of these two different classes corresponding to the greater root of the characteristic equation are locally stable, solutions of the other one corresponding to the smaller root are unstable, and the population dies out if the model does not admit separable solutions or if initial densities of newborns are small enough in the case of the existence of separable solutions. In the case of constant vital rates, the model has no separable solutions or admits only one class of such ones that are globally stable.     

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.     

Sustainability and optimal control of an exploited prey predator system through provision of alternative food to predator

In the present paper, we develop a simple two species prey-predator model in which the predator is partially coupled with alternative prey. The aim is to study the consequences of providing additional food to the predator as well as the effects of harvesting efforts applied to both the species. It is observed that the provision of alternative food to predator is not always beneficial to the system. A complete picture of the long run dynamics of the system is discussed based on the effort pair as control parameters. Optimal augmentations of prey and predator biomass at final time have been investigated by optimal control theory. Also the short and large time effects of the application of optimal control have been discussed. Finally, some numerical illustrations are given to verify our analytical results with the help of different sets of parameters.     

On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment

In this paper we extend the deterministic model for the epidemics induced by virulent phages on bacteria in marine environment introduced by Beretta and Kuang [Math. Biosci. 149 (1998) 57], allowing random fluctuations around the positive equilibrium. The stochastic stability properties of the model are investigated both analytically and numerically suggesting that the deterministic model is robust with respect to stochastic perturbations.     

Slow and fast invasion waves in a model of acid-mediated tumour growth

This work is concerned with a reaction-diffusion system that has been proposed as a model to describe acid-mediated cancer invasion. More precisely, we consider the properties of travelling waves that can be supported by such a system, and show that a rich variety of wave propagation dynamics, both fast and slow, is compatible with the model. In particular, asymptotic formulae for admissible wave profiles and bounds on their wave speeds are provided.     

The Glycine Residues G551 and G1349 within the ATP-Binding Cassette Signature Motifs Play Critical Roles in the Activation and Inhibition of Cystic Fibrosis Transmembrane Conductance Regulator Channels by Phloxine B

The cystic fibrosis transmembrane conductance regulator (CFTR) protein contains a canonical ATP-binding cassette (ABC) signature motif, LSGGQ, in nucleotide binding domain 1 (NBD1) and a degenerate LSHGH in NBD2. Here, we studied the contribution of the conserved residues G551 and G1349 to the pharmacological modulation of CFTR chloride channels by phloxine B using iodide efflux and whole-cell patch clamp experiments performed on the following green fluorescent protein (GFP)-tagged CFTR: wild-type, delF508, G551D, G1349D, and G551D/G1349D double mutant. We found that phloxine B stimulates and inhibits channel activity of wild-type CFTR (K_s = 3.2 ± 1.6 μM, K_i = 38 ± 1.4 μM) and delF508 CFTR (K_s = 3 ± 1.8 μM, K_i = 33 ± 1 μM). However, CFTR channels with the LSGDQ mutated motif (mutation G551D) are activated (K_s = 2 ± 1.13 μM) but not inhibited by phloxine B. Conversely, CFTR channels with the LSHDH mutated motif (mutation G1349D) are inhibited (K_i = 40 ± 1.01 μM) but not activated by phloxine B. Finally, the double mutant G551D/G1349D CFTR failed to respond not only to phloxine B stimulation but also to phloxine B inhibition, confirming the importance of both amino acid locations. Similar results were obtained with genistein, and kinetic parameters were determined to compare the pharmacological effects of both agents. These data show that G551 and G1349 control the inhibition and activation of CFTR by these agents, suggesting functional nonequivalence of the signature motifs of NBD in the ABC transporter CFTR.     

Nanoscale organization and dynamics of the siglec CD22 cooperate with the cytoskeleton in restraining BCR signalling

Receptor organization and dynamics at the cell membrane are important factors of signal transduction regulation. Using super‐resolution microscopy and single‐particle tracking, we show how the negative coreceptor CD22 works with the cortical cytoskeleton in restraining BCR signalling. In naïve B cells, we found endogenous CD22 to be highly mobile and organized into nanodomains. The landscape of CD22 and its lateral diffusion were perturbed either in the absence of CD45 or when the CD22 lectin domain was mutated. To understand how a relatively low number of CD22 molecules can keep BCR signalling in check, we generated Brownian dynamic simulations and supported them with ex vivo experiments. This combined approach suggests that the inhibitory function of CD22 is influenced by its nanoscale organization and is ensured by its fast diffusion enabling a “global BCR surveillance” at the plasma membrane.     

The effects of harvesting of the South African kelp (Ecklonia maxima) on kelp population structure, growth rate and recruitment

Ecklonia maxima is an economically important kelp in South Africa. The harvested kelp is used mainly as feed for cultured Haliotis midae (abalone) on farms all along the South African South and West Coast. The effects that different harvesting methods have on the growth of sub-canopy kelps, kelp population structure and kelp recruitment were tested in a kelp bed at Bordjies Rif near Cape Town. Two 30 × 10 m sites were set up, about 100 m apart, in near monoculture stands of E. maxima. Each 30×10 m area was subdivided into three treatments. In treatment 1 (T1) the whole ‘head’ of each kelp sporophyte that reached the surface was cut off between the bulb and the primary blade (‘lethal’ method). In treatment 2 (T2) (‘non-lethal’ method), the secondary fronds of all sporophytes that reached the surface were cut 20–30 cm from their junction with the primary blade, and removed. In the control plot, the kelp plants were not treated. Harvesting treatments were done approximately every four months, at low spring tide, from 3 March 2003 to 3 November 2003 (three treatments). The effects of harvesting on the kelps depend largely on the size of plant and the time the fronds were removed; however, no seasonal pattern could be observed. The different treatments had no effect on the growth rate, population structure or recruitment of the kelp. This means that factors other than light play an important role in the growth, structure and recruitment of the kelp beds in False Bay. Results are discussed in relation to current commercial harvesting practices.     

The mixed lymphocyte reaction: selective activation and inactivation of the stimulating cells.

Allogeneic cells pretreated for 48 hr with 2 × 10^?6M ouabain have lost the capacity to show the mixed lymphocyte reaction (MLR). Analysis of various combinations of cells in the one-way MLR revealed that this effect was on the stimulating cells and not on the responding cells. Pretreatment of cells from both donors with 10^?7M ouabain caused no change in incorporation of labeled thymidine into DNA during the first 5 days of mixed lymphocyte culture; thereafter, as incorporation by the controls declined, that of the pretreated cells continued to increase. This effect was also on the stimulating cells and not on the responders. The irreversible effects of ouabain are thus either to activate or inactivate the stimulating cells depending on the concentration of the drug; there is little or no effect on the responding cells.     

西藏野生芥菜型油菜构件种群相互关系及其与气候因素的灰色关联度分析

以生物构件理论为基础,运用灰色关联分析技术对西藏野生芥菜型油菜种群构件结构与环境因子的关系进行了研究。结果表明:1)每株角果数、每角果粒数和粒重是产量构成的重要指标,随着每株角果数、每角果粒数和粒重的增大,西藏野生芥菜型油菜的产量明显增加,它们之间的构件因子关联度较大;2)主茎系统是西藏野生芥菜型油菜植株的基础,其间的关联性显示了西藏野生芥菜型油菜植株构件组成的整合作用;3)一级分枝长度、一级分枝角果数、二级分枝数、一级分枝粗度之间有较大的关联度,但与一级分枝发生高度关联度相对较小;4)研究的7个环境因子与西藏野生芥菜型油菜种群的生长关系均较密切,且温度是影响西藏野生芥菜型油菜分布和生命活动相对重要的环境因子,而降水为次要因子。     

Crystal structure of the vitamin D nuclear receptor ligand binding domain in complex with a locked side chain analog of calcitriol

The crystal structures of vitamin D nuclear receptor (VDR) have revealed that all compounds are anchored by the same residues to the ligand binding pocket (LBP). Based on this observation, a synthetic analog with a locked side chain (21-nor-calcitriol-20(22),23-diyne) has been synthesized in order to gain in entropy energy with a predefined active side chain conformation. The crystal structure of VDR LBD bound to this locked side chain analogue while confirming the docking provides a structural basis for the activity of this compound.     

秦岭北坡雉类种群密度和群落结构的初步研究

本研究将秦岭北坡周至自然保护区划分为4个生态地理类型带,在以Gates截线法调查雉类种群密度的基础上,对不同生态地理类型雉类群落结构进行了研究。结果表明,秦岭北坡雉类的物种多样性与植被和海拔有一定关系,环境的稳定性对雉类物种多样性有着一定影响。调查中还发现红腹锦鸡已分布到海拔2700 m的山坡上。     

pK(a) values for side-chain carboxyl groups of a PGB1 variant explain salt and pH-dependent stability

Determination of pK(a) values of titrating residues in proteins provides a direct means of studying electrostatic coupling as well as pH-dependent stability. The B1 domain of protein G provides an excellent model system for such investigations. In this work, we analyze the observed pK(a) values of all carboxyl groups in a variant of PGB1 (T2Q, N8D, N37D) at low and high ionic strength as determined using (1)H-(13)C heteronuclear NMR in a structural context. The pK(a) values are used to calculate the pH-dependent stability in low and high salt and to investigate electrostatic coupling in the system. The observed pK(a) values can explain the pH dependence of protein stability but require pK(a) shifts relative to model values in the unfolded state, consistent with persistent residual structure in the denatured state. In particular, we find that most of the deviations from the expected random coil values can be explained by a significantly upshifted pK(a) value. We show also that (13)C backbone carbonyl data can be used to study electrostatic coupling in proteins and provide specific information on hydrogen bonding and electrostatic potential at nontitrating sites.     

The supramolecular structure of the GPCR rhodopsin in solution and native disc membranes

Rhodopsin, the prototypical G-protein-coupled receptor, which is densely packed in the disc membranes of rod outer segments, was proposed to function as a monomer. However, a growing body of evidence indicates dimerization and oligomerization of numerous G-protein-coupled receptors, and atomic force microscopy images revealed rows of rhodopsin dimers in murine disc membranes. In this work we demonstrate by electron microscopy of negatively stained samples, blue native- and sodium dodecyl sulphate-polyacrylamide gel electrophoresis, chemical crosslinking, and by proteolysis that native bovine rhodopsin exists mainly as dimers and higher oligomers. These results corroborate the recent findings from atomic force microscopy and molecular modeling on the supramolecular structure and packing arrangement of murine rhodopsin dimers.     

Crystal structure of the C-terminal domain of Mu phage central spike and functions of bound calcium ion

Bacteriophage Mu, which has a contractile tail, is one of the most famous genus of Myoviridae. It has a wide host range and is thought to contribute to horizontal gene transfer. The Myoviridae infection process is initiated by adhesion to the host surface. The phage then penetrates the host cell membrane using its tail to inject its genetic material into the host. In this penetration process, Myoviridae phages are proposed to puncture the membrane of the host cell using a central spike located beneath its baseplate. The central spike of the Mu phage is thought to be composed of gene 45 product (gp45), which has a significant sequence homology with the central spike of P2 phage (gpV). We determined the crystal structure of shortened Mu gp45Δ1-91 (Arg92–Gln197) at 1.5 Å resolution and showed that Mu gp45 is a needlelike structure that punctures the membrane. The apex of Mu gp45 and that of P2 gpV contained iron, chloride, and calcium ions. Although the C-terminal domain of Mu gp45 was sufficient for binding to the E. coli membrane, a mutant D188A, in which the Asp amino acid residue that coordinates the calcium ion was replaced by Ala, did not exhibit a propensity to bind to the membrane. Therefore, we concluded that calcium ion played an important role in interaction with the host cell membrane.     

薇甘菊的生物防治及其天敌在中国的新发现

薇甘菊(Mikania mixeantha H.B.K.)为原产中南美洲的菊科假泽兰属攀缘草本植物,现在广泛分布于东南亚地区并对我国南方部分地区的农林业生产造成严重危害。由于施用化学除草剂会对环境造成污染,所以生物防治更为可靠。大量工作表明,一些真菌和昆虫有望被用来进行薇甘菊的生物控制。本文中报道的薇甘菊天敌小蓑蛾(Acanthopsyche sp.)为国内首次发现,为薇甘菊的生物防治提供了新的线索和启示。