Characterizing the symmetric equilibrium of multi-strain host-pathogen systems in the presence of cross immunity

We investigate the population dynamics of host-pathogen systems in which the pathogen has a potentially arbitrary number of antigenically distinct strains interacting via cross-immunity. The interior equilibrium configuration of the symmetric multiple strain SIR model with cross-immunity is characterized. We develop an efficient iterative method for numerically solving the equilibrium equation together with a number of informative analytical approximations to the full solution. Equilibrium properties are studied as a function of the number of strains, reproduction number, infectious period, and cross immunity profile. We establish that the prevalence in the system increases monotonically with the number of strains and the reduction in cross immunity. Moreover, we demonstrate the existence of a phase transition separating high prevalence and low prevalence parameter regions, with the critical point being defined by R₀1, where is the level of cross-immunity and R₀ is the reproduction number. Above the threshold, prevalence saturates with increasing numbers of strains as a result of the inclusion of prohibition of co-infection in the model. Below the threshold, prevalence saturates much more rapidly as the number of strains increases - indicating that when cross-protection is sufficiently intense, the selective advantage for a pathogen to increase its diversity is substantially less than in the threshold region. Similarly, there is limited benefit to increased transmissibility (or decreased cross-immunity) both for the high and low diversity pathogen systems compared with systems at the threshold R₀1 where small increase in transmissibility can result in significant increase in prevalence.     

Thresholds for macroparasite infections

We analyse here the equilibria of an infinite system of partial differential equations modelling the dynamics of a population infected by macroparasites. We find that it is possible to define a reproduction number R₀ that satisfies the intuitive definition, and yields a sharp threshold in the behaviour of the system: if R₀ < 1, the parasite-free equilibrium (PFE) is asymptotically stable and there are no endemic equilibria; if R₀ > 1, the PFE is unstable and there exists a unique endemic equilibrium. The results mainly confirm what had been obtained in simplified models, except for the fact that no backward bifurcation occurs in this model. The stability of equilibria is established by extending an abstract linearization principle and by analysing the spectra of appropriate operators.Revised version: 14 November 2003Supported in part by CNR under Grant n. 00.0142.ST74 Metodi e modelli matematici nello studio dei fenomeni biologici     

Analysis of a disease transmission model in a population with varying size

An S I R S epidemiological model with vital dynamics in a population of varying size is discussed. A complete global analysis is given which uses a new result to establish the nonexistence of periodic solutions. Results are discussed in terms of three explicit threshold parameters which respectively govern the increase of the total population, the existence and stability of an endemic proportion equilibrium and the growth of the infective population. These lead to two distinct concepts of disease eradication which involve the total number of infectives and their proportion in the population.Partially supported by NSF Grant No. DMS-8703631. This work was done while this author was visiting the University of VictoriaResearch supported in part by NSERC A-8965     

A neural net capable of competitive and cooperative computation

An algorithm of learning in multilayer threshold nets without feedbacks is proposed. The net is. built of threshold elements with binary inputs. During a learning process each input vector x is accompanied by a teacher's decision ({1,...,M}). The pairs (x[n], [n]) appear in successive steps independently according to some unknown stationary distribution p(x,). The problem of learning of a threshold net has been decomposed to a series of problems of learning of the threshold elements. The proposed learning algorithm of the threshold elements has a perceptron-like form. It was proven that a decision rule of the threshold net stabilizes after a finite number of steps. For definite classes {p(x, )} _* ^K of distributions p(x,), an optimal decision rule stabilizes after a finite number of steps. These classes {p(x, )} _* ^K also contain distributions describing learning processes with perturbations.     

A generalized model of a resource-population system

Summary A system of nonlinear differential equations describing a resource-population system is analyzed in terms of the existence and characteristics of its equilibrium states.It is proved that, under the condition that k< (necessary condition for the population being able to grow under optimal conditions), it is a necessary and sufficient condition for the system to have a steady state that the resource input rate to the system be constant.When the resource input rate is a constant different from zero, the system has only one equilibrium point, at M ⁰=/⁰/k, A ⁰=–(/⁰/k)ln(1–k/), and this equilibrium point is always stable. In other words, the system population-resource will always reach the steady state, either monotonically (node) or by damped oscillations (focus), from any arbitrary initial condition in the positive quadrant.When the resource input rate is equal to zero, the system has an infinite number of equilibrium points at M ₀=0, A ₀=constant. All these equilibrium points are unstable in the sense that any slight increase in M will move the system away from the equilibrium states, except for the point M ₀=0, A ₀=0, which is the only stable equilibrium point, to which the system will tend. This stable equilibrium point corresponds to the condition of complete annihilation of both resource and population.Finally, it is proved that the system does not have limit cycles in the positive quadrant and is therefore incapable of self-oscillations.This work was partially supported by a Ford Foundation fellowship and various Cornell University fellowships.     

The Hawk—Dove game revisited: Effects of continuous variation in resource-holding potential on the frequency of escalation

Summary The classic Hawk—Dove game is extended to deal with continuous variation in resource-holding potential or RHP, when RHP is observable (via any sensory modality) but RHP difference is less than perfectly reliable as a predictor of the outcome of an escalated contest. The relationship between sensory and physical magnitudes of RHP is assumed to be governed by Fechner's psychophysical law, whose effect is that contestants interact as if they had perfect information about their relative RHP (as opposed to RHP difference). Thus, an animal is aggressive if its RHP exceeds a certain fraction, called its threshold, of its opponent's RHP and otherwise is non-aggressive; and the classic Hawk and Dove strategies correspond to zero and infinite thresholds, respectively. For RHPs drawn at random from an arbitrary Gamma distribution there is a unique evolutionarily stable strategy or ESS, which depends on a parameter measuring the reliability of RHP as a predictor of the outcome of a fight, on the ratio of the valueV of winning to the costC of losing (both measured in units of reproductive fitness) and on the mean µ and variance ² of the RHP distribution. In a population at this ESS, ifV/C < 1 then the threshold is 1 and there is no fighting. AsV/C increases beyond 1 to a second critical value , however, the threshold decreases steadily from 1 to 0 and remains 0 forV/C > ; is an increasing function of , but a decreasing function of ². That a lower variance of RHP can imply a lower escalation frequencyp is a novel insight of the analysis. The prediction is at first counterintuitive, because if the aggression threshold were fixed then larger variance would imply lowerp (dispersion effect of variance). When natural selection acts on the threshold, however, increasing the variance not only reduces the probability that an animal with larger RHP will be attacked by an animal with lower RHP at the existing threshold, but also reduces the expected costs of adopting that particular threshold, so that a mutant with a somewhat lower threshold can invade the population (selection effect of variance). Forp, the selection effect dominates toward the upper end of the interval 1 V/C .     

Relationship between α-L-fucosidase deficiency in plasma and α-L-fucosidase activity in leukocytes

Summary After testing a population sample of 185 hospitalized Italian children for the plasma -L-fucosidase deficiency and establishing an approximate threshold value between heterozygotes and wild-type homozygotes, we analyzed by two statistical methods the distribution of the two genotypes. The results obtained by probit analysis agree with threshold and average values expected on the basis of the Hardy-Weinberg equilibrium.In addition, the level of -fucosidase in leukocytes of 12 individuals with deficiency of -fucosidase in plasma was found to be significantly lower than that of 61 controls (P<0.005). These results indicate that the mutation(s) causing a deficiency of -fucosidase in plasma is (are) also expressed in leukocytes.     

Integration time for short broad band clicks in echolocating FM-bats (Eptesicus fuscus)

Vespertilionid FM-bats (four Eptesicus fuscus and one Vespertilio murinus) were trained in an electronic phantom target simulator to detect synthetic echoes consisting of either one or two clicks. The threshold sound pressure for single clicks was around 47 dB peSPL for all five bats corresponding to a threshold energy of -95 dB re 1 Pa² * s. By varying the interclick interval, T, for double clicks it was shown that the threshold intensity was around — 3 dB relative to the threshold for single clicks at T up to 2.4 ms, indicating perfect power summation of both clicks. A threshold shift of -13.5 dB for a 1 ms train of 20 clicks (0.05 ms interclick interval) confirmed that the bats integrated the power of the stimuli. At T longer than around 2.5 ms the threshold for double clicks was the same as for single clicks. Thus, the bats performed like perfect energy detectors with an integration time of approximately 2.4 ms. This integration time is an order of magnitude shorter than that reported for bats listening passively for pure tones. In our setup the bats emitted sonar signals with durations of 2–3 ms. Hence, the results may indicate that while echolocating the bats integration time is adapted to the duration of the sonar emissions.Abbreviations AGC automatic gain control - FM frequency modulated - peSPL peak equivalent sound pressure level - rms root mean square - SD standard deviation - SE standard error of mean - T interclick interval     

Midday depression in net photosynthesis and stomatal conductance in Yucca glauca

Summary Yucca glauca in the Colorado shortgrass prairie undergoes a pronounced midday depression in net photosynthesis and stomatal conductance under summer field conditions. This phenomenon can be duplicated in the laboratory using potted plants by simulating a typical summer daily pattern of leaf temperature and leaf-to-air water vapor concentration difference (w). The decrease in photosynthetic rate appears to be due primarily to high leaf temperatures, while the decrease in stomatal conductance can be attributed mainly to high w values. Stomatal conductance also decreases when leaf temperatures exceed a critical threshold value, even when w is artificially maintained at a constant level. The threshold temperature is commonly attained for leaves in situ, but only after substantial stomatal closure has already occurred as a result of high w values.The photosynthetic temperature optimum and threshold temperature which promotes stomatal closure increases substantially as the growing season progresses. As a result, the midday depression in photosynthesis occurs at higher temperatures in mid-summer than in late spring. Preliminary evidence suggests that the photosynthetic temperature optimum closely follows the naturally-occurring morning leaf temperatures, while the threshold temperature for stomatal closure matches afternoon leaf temperatures.     

Bioenergetic characterization of transient state phosphate uptake by the cyanobacterium Anacystis nidulans

A force flow relationship based on nonequilibrium thermodynamics was derived to analyze the variable transient state phosphate uptake phenomena of cyanobacteria seen under different growth conditions and external phosphate concentrations. This relationship postulates the following basic properties of the uptake system: First, a threshold value exists, below which incorporation is energetically impossible. Second, threshold values are influenced by the activity of the phosphate uptake system, such that a decrease of the activity increases the threshold level. Third, near the thermodynamic equilibrium the uptake rate is linearly dependent on the free energy of polyphosphate formation and the pH-gradient at the thylakoid membrane. Experiments performed with Anacystis nidulans showed that phosphate uptake characteristics conformed to the properties predicted by the linear force-flow relationship. Linearity extented into regions far form thermodynamic equilibrium, e.g. to high phosphate concentrations, when algae were preconditioned to high phosphate levels. Under phosphate limited growth linearity was confined to a small concentration range, threshold values decreased below 10 nM, and the external concentration approached threshold. The data suggest that the uptake system responds to changes in the external phosphate concentration in the same way as sensory systems to input stimuli by amplifying signals and adapting to them.Abbreviations chl chlorophyll - H _e ⁺ , H _C ⁺ , H _T ⁺ protons in the external medium, the cytoplasmic and thylakoid space respectively - P_c phosphate in the cytoplasmic space - P_e phosphate in the external medium - P_n, P_n+1 polyphosphates - pH_T pH-gradient across the thylakoid membrane     

Sensitivity analysis of equilibrium in density-dependent matrix population models

We consider the effects of parameter perturbations on a density‐dependent population at equilibrium. Such perturbations change the dominant eigenvalue λ of the projection matrix evaluated at the equilibrium as well as the equilibrium itself. We show that, regardless of the functional form of density dependence, the sensitivity of λ is equal to the sensitivity of an effective equilibrium density , which is a weighted combination of the equilibrium stage densities. The weights measure the contributions of each stage to density dependence and their effects on demography. Thus, is in general more relevant than total density, which simply adds all stages regardless of their ecological properties. As log λ is the invasion exponent, our results show that successful invasion will increase , and that an evolutionary stable strategy will maximize . Our results imply that eigenvalue sensitivity analysis of a population projection matrix that is evaluated near equilibrium can give useful information about the sensitivity of the equilibrium population, even if no data on density dependence are available.     

The influence of amino-reactive substances on contraction threshold of frog skeletal muscle

Summary The action of the amino-reactive substances pyridoxal phosphate, 4-acetamido-4-isothiocyanato-stilbene-2,2-disulfonic acid and 2,4,6-trinitrobenzene sulfonic acid on the contraction threshold, taken as parameter for the initiation of contraction, was investigated in fibers of the sartorius muscle of the frog. The contraction threshold was shifted by 1 to 11 mV tomore negative potentials with 1 to 20mm PDP. Similar shifts from 2 to 17 mV were produced by 0.66 to 20mm SITS. The threshold shift was only partially reversible. The shift of the contraction threshold obtained with 2mm SITS was nearly constant at different [Ca²⁺]_o and [Mg²⁺]_o from 1.5 to 50mm with a tendency to increase at higher divalent cation concentration. TNBS had no effect on the contraction threshold.The action of PDP and SITS on the contraction threshold was successfully described by the surface charge model used earlier to explain the effect of lanthanum, neuraminidase and ruthenium red on the contraction threshold (M. Dörrscheidt-Käfer,Pfluegers Arch. 380:171–179, 181–187, 1979;J. Membrane Biol. 62:95–103, 1981). Here it was assumed that PDP and SITS bind to positive fixed charges on the surface of the T-tubular wall. This results in a shift of the calculated surface potential to more negative values which is thought to account for the measured shift of the contraction threshold.     

Function of the hemoglobin and the gas bubble in the backswimmerAnisops assimilis (Hemiptera: Notonectidae)

Summary The presence of hemoglobin inAnisops assimilis has been demonstrated to be a vital factor in the physiology of this organism. The hemoglobin is composed of heterogeneous subunits which aggregate upon deoxygenation. This association-dissociation equilibrium confers a steep gradient (n _H6) to the oxygen equilibrium curve and a low oxygen affinity (P ₅₀40 mmHg). Oxygen bound by the hemoglobin is released into a gas bubble enabling the bug to regulate its density around that of water. Thus, energy is conserved during a dive, allowing the animal to remain in mid-water for long periods. This adaptive feature has facilitated the exploitation of an ecological niche available to few other organisms.     

Anaerobic metabolism of goldfish,Carassius auratus (L.). Influence of anoxia on mass-action ratios of transaminase reactions and levels of ammonia and succinate

Summary Changes in the concentrations of ammonia, glutamate, alanine, aspartate, -ketoglutarate, oxaloacetate and succinate were measured in freeze-clamped lateralred muscle, dorsal white muscle and liver, and in rapidly cooled blood of goldfish after 12 h of anoxia. Alanine accumulation, succinate accumulation and aspartate depletion are observed in all tissues examined; in the liver the concentrations of glutamate increase and those of ammonia decrease. The mass-action ratio of the glutamate-pyruvate transaminase-catalyzed reaction stays within one order of magnitude from thermodynamic equilibrium in the direction of alanine formation. The mass-action ratio of the glutamate-oxaloacetate transaminase reaction is far from equilibrium when measured oxaloacetate concentrations are used. When levels of free oxaloacetate are calculated from LDH and MDH equilibrium constants, the mass-action ratio of glutamate-oxaloacetate transamination is close to equilibrium in the direction of aspartate formation. Since neither alanine nor glutamate decreases, and since ammonia gradients suggest a continuous ammonia production in all tissues examined, anaerobic proteolysis is assumed. A possible coupling between amino acid catabolism and ethanol production is discussed.Abbreviations ALA alanine - ASP aspartate - EDTA ethylene diamine tetraacetate - FP _ox oxidated flavoprotein - FP _red reduced flavoprotein - FUM fumarate - GLU glutamate - GOT glutamate oxaloacetate transaminase - GPT glutamate pyruvate transaminase - IMP inosine monophosphate - KG -ketoglutarate - LDH lactate dehydrogenase - MAL malate - MAR mass action ratio - MDH malate dehydrogenase - OAA oxaloacetate - PYR pyruvate - sAMP adenylosuccinate - SDH succinate dehydrogenase - SUCC succinate     

A gradually slowing travelling band of chemotactic bacteria

A model for describing the motion of chemotactic bacteria in a capillary tube containing substrate is treated. Chemotactic substrate threshold effects are included in the chemotactic response coefficient. The ratio of the substrate threshold, s _T, to the substrate level far ahead of the travelling band, s , is used as a small parameter in developing an asymptotic solution of near travelling wave form.Also at: Membrane Filtration Technology, Kiryat Weizmann, Rehovot, Israel.Also at: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12181, USA.     

A simple phenomenological thermodynamic model for protein partitioning in reversed micellar systems

A simple thermodynamic model is developed for the partitioning of proteins between a bulk aqueous solution and a reversed micellar organic phase by assuming that a pseudo-chemical equilibrium is established when proteins in solution interact with a non-integral number of empty micelles to form the protein-micelle complex. From the equilibrium constant for this reaction, which is related to both the chemical and electrical free energy changes associated with the transfer of the proteins between the two phases, a simple expression is derived for the partition coefficient as a function of pH and surfactant concentration. Assumptions include a linear variation in protein net charge with pH, and a linear decrease in protein-micelle complex size with increasing protein charge. Results on the solubilization of ribonuclease-a and concanavalin-a in Aerosol-OT/isooctane organic solvents were well-correlated by the model equation, and the estimated parameters were of the expected order of magnitude as estimated based on the known physical properties of the system components.List of Symbols F C/mol Faraday's constant - G J/mol standard free energy change on solubilization - G J/mol standard free energy change in the absence of charge effects - K partition coefficient - K _eq (mol/m³)^–n equilibrium constant for pseudo-reaction (1) - M micelle - N _ag empty micelle aggregation number - n number of empty micelles required to form protein/micelle complex - n ₀ number of empty micelles required at zero net protein charge - P protein - PM protein/micelle complex - pI protein isoelectric point - R J/mol K gas law constant - S surfactant - z protein charge - slope of protein titration curve - change in micelle size, n, per unit change in charge - V electrostatic potential difference     

A neural coding model for sensory intensity discrimination,to be applied to gustation

Summary This paper presents a model of the neural coding and discrimination of sensory intensity. The model consists of five stages: (1) the coding of stimulus intensity in peripheral receptors or neurons by a rate code. The relevance of comparing different analysis intervals for the response is pointed out; (2) neural processing, according to either labeled-line or across-fiber pattern theory. In addition, two possible non-linearities in the processing are considered: a threshold mechanism, and contrast enhancement by reciprocal inhibition; (3) a neural discriminator, based on signal-detection theory; (4) a memory stage; (5) an effector organ providing a behavioral output. Emphasis is put on stages 2 and 3.The model produces predictions of the differential threshold, which should be directly testable in a behavioral two-alternative forced-choice paradigm. The model will be applied to gustatory intensity discrimination in rat in a subsequent study (Maes and Erickson 1984). The Discussion pays attention to the relative contributions of peripheral and central noise sources. It also compares the present model with Beidler's (1958) approach through just noticeable differences (JND's). The model presented here seems more adequate in providing an understanding of sensory information processing.Abbreviations AFP across fiber pattern - DA discrimination acuity - DT differential threshold - JND just noticeable difference - LL labeled line - NTS nucleus tractus solitarius     

Evolution under the multilocus Levene model without epistasis

Evolution under the multilocus Levene model is investigated. The linkage map is arbitrary, but epistasis is absent. The geometric-mean fitness, , depends only on the vector of gene frequencies, ρ; it is nondecreasing, and the single-generation change is zero only on the set, Λ, of gametic frequencies at gene-frequency equilibrium. The internal gene-frequency equilibria are the stationary points of . If the equilibrium points of ρ(t) (where t denotes time in generations) are isolated, as is generic, then ρ(t) converges as t→∞ to some . Generically, ρ(t) converges to a local maximum of . Write the vector of gametic frequencies, p, as , where d represents the vector of linkage disequilibria. If is a local maximum of , then the equilibrium point is asymptotically stable. If either there are only two loci or there is no dominance, then d(t)→0 globally as t→∞. In the second case, has a unique maximum and is globally asymptotically stable. If underdominance and overdominance are excluded, and if at each locus, the degree of dominance is deme independent for every pair of alleles, then the following results also hold. There exists exactly one stable gene-frequency equilibrium (point or manifold), and it is globally attracting. If an internal gene-frequency equilibrium exists, it is globally asymptotically stable. On Λ, (i) the number of demes, Γ, is a generic upper bound on the number of alleles present per locus; and (ii) if every locus is diallelic, generically at most Γ−1 loci can segregate. Finally, if migration and selection are completely arbitrary except that the latter is uniform (i.e., deme independent), then every uniform selection equilibrium is a migration-selection equilibrium and generically has the same stability as under pure selection.     

Kinetic and Equilibrium Studies on d-Mannose-d-Fructose Isomerization Catalyzed by Mannose Isomerase from Streptomyces aerocolorigenes

The equilibrium constant of the isomerization reaction between d-mannose and d-fructose which is catalyzed by a mannose isomerase from Streptomyces aerocolorigenes was obtained by using three methods over the temperature range from 1 to 40°C.

It was found that the equilibrium constant was scarcely dependent on temperature, ΔH, the heat of the formation of d-fructose from d-mannose, being approximately zero.

The standard free energy change, ΔG, and the standard entropy change, ΔS, of the reaction were calculated from the equilibrium constants at various temperatures and ΔH. The values of ΔG and ΔS at 25°C were ?650 cal/mole and + 2.2 cal/deg·mole, respectively.

By combining these thermodynamic data with those obtained for the isomerization reaction between d-glucose and d-fructose reported in the previous paper, ΔH, ΔG and ΔS for the isomerization between d-mannose and d-glucose were indirectly obtained to be +2220 cal/mole, +830 cal/mole and +4.6 cal/deg·mole at 25°C, respectively.     

Interaction ofH-2 and nonH-2 linked genes in the regulation of antibody response to a threshold dose of sheep erythrocytes

The antibody response to a threshold dose (¹⁰) of SE was studied in the High responder line (H) and the Low responder line (L) of mice obtained by bidirectional selective breeding for the character quantitative agglutinin response to an optimal dose of SE, and in interline hybrids: F₁, F₁ and both backcrosses. Whereas the interline difference in agglutinin responses at the optimal dose is due to the additive effect of about ten independently segregating loci, one of which isH-2 linked, the responsiveness to the threshold dose is determined by the effect of two loci. The direction of the dominance effect also varies with the antigen dose: high responsiveness is partially dominant at the optimal dose while at the threshold dose nonresponder character is partially dominant. The role of theH-2 linked locus was investigated. It has been demonstrated that on an identical background (equivalent to that of F₁ hybrids) this locus is responsible for 12% of the interline difference at the optimal antigen dose, and for 61% at the threshold antigen dose. For the two antigen doses, the quantitative effect of theH-2 locus is in agreement with the estimate of the number of loci obtained by variance analysis. The intervention of a second gene, non-H-2 linked, in the regulation of responsiveness to 10⁶ SE is demonstrated by appropriate assortative matings. The interaction between the two genes is discussed.