Stochastic epidemics in dynamic populations: quasi-stationarity and extinction

Empirical evidence shows that childhood diseases persist in large communities whereas in smaller communities the epidemic goes extinct (and is later reintroduced by immigration). The present paper treats a stochastic model describing the spread of an infectious disease giving life-long immunity, in a community where individuals die and new individuals are born. The time to extinction of the disease starting in quasi-stationarity (conditional on non-extinction) is exponentially distributed. As the population size grows the epidemic process converges to a diffusion process. Properties of the limiting diffusion are used to obtain an approximate expression for τ, the mean-parameter in the exponential distribution of the time to extinction for the finite population. The expression is used to study how τ depends on the community size but also on certain properties of the disease/community: the basic reproduction number and the means and variances of the latency period, infectious period and life-length. Effects of introducing a vaccination program are also discussed as is the notion of the critical community size, defined as the size which distinguishes between the two qualitatively different behaviours. Received: 14 February 2000 / Revised version: 5 June 2000 / Published online: 24 November 2000     

Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system

The recent broad interest on ratio-dependent based predator functional response calls for detailed qualitative study on ratio-dependent predator-prey differential systems. A first such attempt is documented in the recent work of Kuang and Beretta(1998), where Michaelis-Menten-type ratio-dependent model is studied systematically. Their paper, while contains many new and significant results, is far from complete in answering the many subtle mathematical questions on the global qualitative behavior of solutions of the model. Indeed, many of such important open questions are mentioned in the discussion section of their paper. Through a simple change of variable, we transform the Michaelis-Menten-type ratio-dependent model to a better studied Gause-type predator-prey system. As a result, we can obtain a complete classification of the asymptotic behavior of the solutions of the Michaelis-Menten-type ratio-dependent model. In some cases we can determine how the outcomes depend on the initial conditions. In particular, open questions on the global stability of all equilibria in various cases and the uniqueness of limit cycles are resolved. Biological implications of our results are also presented.     

A unified hypothesis of coeliac disease with implications for management of patients

Summary. This mini-review presents the research carried out within the context of two of the main hypotheses of the aetiology of coeliac disease. The enzymopathic hypothesis of the disease has been placed clearly as the underlying deficiency causing increased levels of toxic peptides, while the immunological hypothesis has been implicated in the pathogenesis of the disorder as the result of the action of undigested peptides in the small intestine. As a consequence, we are proposing a unified hypothesis of coeliac disease, which takes into account the actions of these undigested peptides through their direct cytotoxicity and their immunoactivity. At the same time, work aimed at defining some of these biologically active peptides, which could be said to be involved in the aetiopathogenesis of coeliac disease, will be reported. The review also focusses on the use of enzyme therapy for management of the disease, which when used in conjunction with the gluten-free diet, offers a safeguard against damage to the small intestine caused by small amounts of gluten.     

Stochastic models of a parasitic infection, exhibiting three basic reproduction ratios

Two closely related stochastic models of parasitic infection are investigated: a non-linear model, where density dependent constraints are included, and a linear model appropriate to the initial behaviour of an epidemic. Host-mortality is included in both models. These models are appropriate to transmission between homogeneously mixing hosts, where the amount of infection which is transferred from one host to another at a single contact depends on the number of parasites in the infecting host. In both models, the basic reproduction ratio R0 can be defined to be the lifetime expected number of offspring of an adult parasite under ideal conditions, but it does not necessarily contain the information needed to separate growth from extinction of infection. In fact we find three regions for a certain parameter where different combinations of parameters determine the behavior of the models. The proofs involve martingale and coupling methods.     

An explicit representation of the Luria–Delbrück distribution

The probability distribution of the number of mutant cells in a growing single-cell population is presented in explicit form. We use a discrete model for mutation and population growth which in the limit of large cell numbers and small mutation rates reduces to certain classical models of the Luria-Delbrück distribution. Our results hold for arbitrarily large values of the mutation rate and for cell populations of arbitrary size. We discuss the influence of cell death on fluctuation experiments and investigate a version of our model that accounts for the possibility that both daughter cells of a non-mutant cell might be mutants. An algorithm is presented for the quick calculation of the distribution. Then, we focus on the derivation of two essentially different limit laws, the first of which applies if the population size tends to infinity while the mutation rate tends to zero such that the product of mutation rate times population size converges. The second limit law emerges after a suitable rescaling of the distribution of non-mutant cells in the population and applies if the product of mutation rate times population size tends to infinity. We discuss the distribution of mutation events for arbitrary values of the mutation rate and cell populations of arbitrary size, and, finally, consider limit laws for this distribution with respect to the behavior of the product of mutation rate times population size. Thus, the present paper substantially extends results due to Lea and Coulson (1949), Bartlett (1955), Stewart et al. (1990), and others.     

The influence of drug treatment on the maintenance of schistosome genetic diversity

Drug treatment of patients with schistosomiasis may select for drug-resistant parasites. In this article, we formulate a deterministic model with multiple strains of schistosomes (helminth parasites with a two-host life cycles) in order to explore the role of drug treatment in the maintenance of a polymorphism of parasite strains that differ in their resistance levels. The basic reproductive numbers for all strains are computed, and are shown to determine the stabilities of equilibria of the model and consequently the distribution of parasite phenotypes with different levels of drug tolerance. Analysis of our model shows that the likelihood that resistant strains will increase in frequency depends on the interplay between their relative fitness, the cost of resistance, and the degree of selection pressure exerted by the drug treatments.     

A mathematical model of the stress induced during avascular tumour growth

In this paper a mathematical model is developed to describe the effect of nonuniform growth on the mechanical stress experienced by cells within an avascular tumour. The constitutive law combines the stress-strain relation of linear elasticity with a growth term that is derived by analogy with thermal expansion. To accommodate the continuous nature of the growth process, the law relates the rate of change of the stress tensor to the rate of change of the strain (rather than relating the stress to the strain directly). By studying three model problems which differ in detail, certain characteristic features are identified. First, cells near the tumour boundary, where nutrient levels and cell proliferation rates are high, are under compression. By contrast, cells towards the centre of the tumour, where nutrient levels are low and cell death dominant, are under tension. The implications of these results and possible model developments are also discussed. Received: 15 November 1999 / Published online: 5 May 2000     

Lipid nitration and formation of lipid-protein adducts: biological insights

Summary. Lipid-protein adducts are formed during oxidative and nitrative stress conditions associated with increasing lipid and protein oxidation and nitration. The focus of this review is the analysis of interactions between oxidative-modified lipids and proteins and how lipid nitration can modulate lipid-protein adducts formation. For this, two biologically-relevant models will be analysed: a) human low density lipoprotein, whose oxidation is involved in the early steps of atherogenesis, and b) α-synuclein/lipid membranes system, where lipid-protein adducts are being associated with the develop of Parkinson disease and other synucleinopathies.     

Abnormal changes of plasma acute phase proteins in schizophrenia and the relation between schizophrenia and haptoglobin (Hp) gene

Summary. In this study we focused on detecting schizophrenia related changes of plasma proteins using proteomic technology and examining the relation between schizophrenia and haptoglobin (Hp) genotype. We investigated plasma proteins from schizophrenic subjects (n = 42) and healthy controls (n = 46) by two-dimensional gel electrophoresis (2-DE) in combination with mass spectrometry. To further reveal the genetic relationship between acute phase proteins (APPs) and schizophrenia disease, we tested Hp α1/Hp α2 (Hp 1/2) polymorphism and two single nucleotide polymorphisms (SNPs) of Hp, rs2070937 and rs5473, for associations with schizophrenia in the Chinese Han population. With the relatively high number of samples for 2-DE work, we found that four proteins in the family of positive APPs were all up-regulated in patients. In genetic association study, we found significant associations existing between schizophrenia and Hp polymorphisms, Hp 1/2 and rs2070937 variants. Schizophrenia is accompanied by both an altered expression of Hp protein and a different genotype distribution of Hp gene, demonstrating that Hp is associated with schizophrenia. The results from proteomic and genomic aspects both indicate that acute phase reaction is likely to be an aetiological agent in the pathophysiology of schizophrenia, but not just an accompanying symptom. The positive APPs are schizophrenic related proteins, with the highly concordant results on four positive APPs. The first two authors contributed equally.     

Peptide growth factor phytosulfokine-α contributes to the pollen population effect

Density-dependent pollen germination and tube growth in vitro is a well-documented phenomenon, termed the pollen population effect, but far less is known about its molecular basis. We present evidence to support phytosulfokine-α [Y(SO₃H)IY(SO₃H)TQ; PSK-α] as a native bioactive factor contributing to this effect. Mature pollen grains of Nicotiana tabacum L. var.macrophylla were incubated in liquid medium for 2 h. Pollen germination frequency increased in a density-dependent manner from 625 to 46,000 grains/ml. Conditioned medium, obtained from the medium of pollen cultured at a density of 10,000 pollen grains/ml for 12 h, promoted the germination of pollen cultured at a low density (625 grains/ml). A rabbit antiserum against PSK-α specifically inhibited the promotive effect of conditioned medium. Quantification by enzyme-linked immunosorbent assay showed that the conditioned medium contained 0.4 nM of PSK-α. Exogenous PSK-α also stimulated pollen germination in the low-density culture. These results indicate that PSK-α is an important regulator involved in the pollen population effect. Received: 15 March 2000 / Accepted: 24 May 2000     

Evolution of Complex Density-Dependent Dispersal Strategies

The question of how dispersal behavior is adaptive and how it responds to changes in selection pressure is more relevant than ever, as anthropogenic habitat alteration and climate change accelerate around the world. In metapopulation models where local populations are large, and thus local population size is measured in densities, density-dependent dispersal is expected to evolve to a single-threshold strategy, in which individuals stay in patches with local population density smaller than a threshold value and move immediately away from patches with local population density larger than the threshold. Fragmentation tends to convert continuous populations into metapopulations and also to decrease local population sizes. Therefore we analyze a metapopulation model, where each patch can support only a relatively small local population and thus experience demographic stochasticity. We investigated the evolution of density-dependent dispersal, emigration and immigration, in two scenarios: adult and natal dispersal. We show that density-dependent emigration can also evolve to a nonmonotone, “triple-threshold” strategy. This interesting phenomenon results from an interplay between the direct and indirect benefits of dispersal and the costs of dispersal. We also found that, compared to juveniles, dispersing adults may benefit more from density-dependent vs. density-independent dispersal strategies.     

Models of central pattern generators for quadruped locomotion I. Primary gaits

In this paper we continue the analysis of a network of symmetrically coupled cells modeling central pattern generators for quadruped locomotion proposed by Golubitsky, Stewart, Buono, and Collins. By a cell we mean a system of ordinary differential equations and by a coupled cell system we mean a network of identical cells with coupling terms. We have three main results in this paper. First, we show that the proposed network is the simplest one modeling the common quadruped gaits of walk, trot, and pace. In doing so we prove a general theorem classifying spatio-temporal symmetries of periodic solutions to equivariant systems of differential equations. We also specialize this theorem to coupled cell systems. Second, this paper focuses on primary gaits; that is, gaits that are modeled by output signals from the central pattern generator where each cell emits the same waveform along with exact phase shifts between cells. Our previous work showed that the network is capable of producing six primary gaits. Here, we show that under mild assumptions on the cells and the coupling of the network, primary gaits can be produced from Hopf bifurcation by varying only coupling strengths of the network. Third, we discuss the stability of primary gaits and exhibit these solutions by performing numerical simulations using the dimensionless Morris-Lecar equations for the cell dynamics.     

Structured population on two patches: modeling dispersal and delay

We derive from the age-structured model a system of delay differential equations to describe the interaction of spatial dispersal (over two patches) and time delay (arising from the maturation period). Our model analysis shows that varying the immature death rate can alter the behavior of the homogeneous equilibria, leading to transient oscillations around an intermediate equilibrium and complicated dynamics (in the form of the coexistence of possibly stable synchronized periodic oscillations and unstable phase-locked oscillations) near the largest equilibrium.     

Multi-layered network structure of amino acid (AA) metabolism characterized by each essential AA-deficient condition

Summary. The concentrations of free amino acids in plasma change coordinately and their profiles show distinctive features in various physiological conditions; however, their behavior can not always be explained by the conventional flow-based metabolic pathway network. In this study, we have revealed the interrelatedness of the plasma amino acids and inferred their network structure with threshold-test analysis and multilevel-digraph analysis methods using the plasma samples of rats which are fed diet deficient in single essential amino acid. In the inferred network, we could draw some interesting interrelations between plasma amino acids as follows: 1) Lysine is located at the top control level and has effects on almost all of the other plasma amino acids. 2) Threonine plays a role in a hub in the network, which has direct links to the most number of other amino acids. 3) Threonine and methionine are interrelated to each other and form a loop structure.     

Pair of lice lost or parasites regained: the evolutionary history of anthropoid primate lice

Background

Reduction or elimination of vector populations will tend to reduce or eliminate transmission of vector-borne diseases. One potential method for environmentally-friendly, species-specific population control is the Sterile Insect Technique (SIT). SIT has not been widely used against insect disease vectors such as mosquitoes, in part because of various practical difficulties in rearing, sterilization and distribution. Additionally, vector populations with strong density-dependent effects will tend to be resistant to SIT-based control as the population-reducing effect of induced sterility will tend to be offset by reduced density-dependent mortality.

Results

We investigated by mathematical modeling the effect of manipulating the stage of development at which death occurs (lethal phase) in an SIT program against a density-dependence-limited insect population. We found late-acting lethality to be considerably more effective than early-acting lethality. No such strains of a vector insect have been described, so as a proof-of-principle we constructed a strain of the principal vector of the dengue and yellow fever viruses, Aedes (Stegomyia) aegypti, with the necessary properties of dominant, repressible, highly penetrant, late-acting lethality.

Conclusion

Conventional SIT induces early-acting (embryonic) lethality, but genetic methods potentially allow the lethal phase to be tailored to the program. For insects with strong density-dependence, we show that lethality after the density-dependent phase would be a considerable improvement over conventional methods. For density-dependent parameters estimated from field data for Aedes aegypti, the critical release ratio for population elimination is modeled to be 27% to 540% greater for early-acting rather than late-acting lethality. Our success in developing a mosquito strain with the key features that the modeling indicated were desirable demonstrates the feasibility of this approach for improved SIT for disease control.     

Finite metapopulation models with density-dependent migration and stochastic local dynamics

The effects of small density-dependent migration on the dynamics of a metapopulation are studied in a model with stochastic local dynamics. We use a diffusion approximation to study how changes in the migration rate and habitat occupancy affect the rates of local colonization and extinction. If the emigration rate increases or if the immigration rate decreases with local population size, a positive expected rate of change in habitat occupancy is found for a greater range of habitat occupancies than when the migration is density-independent. In contrast, the reverse patterns of density dependence in respective emigration and immigration reduce the range of habitat occupancies where the metapopulation will be viable. This occurs because density-dependent migration strongly influences both the establishment and rescue effects in the local dynamics of metapopulations.     

Two SIS epidemiologic models with delays

 The SIS epidemiologic models have a delay corresponding to the infectious period, and disease-related deaths, so that the population size is variable. The population dynamics structures are either logistic or recruitment with natural deaths. Here the thresholds and equilibria are determined, and stabilities are examined. In a similar SIS model with exponential population dynamics, the delay destabilized the endemic equilibrium and led to periodic solutions. In the model with logistic dynamics, periodic solutions in the infectious fraction can occur as the population approaches extinction for a small set of parameter values. Received: 10 January 1997 / 18 November 1997     

Modelling density-dependent resistance in insect-pathogen interactions.

We consider a mathematical model for a host-pathogen interaction where the host population is split into two categories: those susceptible to disease and those resistant to disease. Since the model was motivated by studies on insect populations, we consider a discrete-time model to reflect the discrete generations which are common among insect species. Whether an individual is born susceptible or resistant to disease depends on the local population levels at the start of each generation. In particular, we are interested in the case where the fraction of resistant individuals in the population increases as the total population increases. This may be seen as a positive feedback mechanism since disease is the only population control imposed upon the system. Moreover, it reflects recent experimental observations from noctuid moth-baculovirus interactions that pathogen resistance may increase with larval density. We find that the inclusion of a resistant class can stabilise unstable host-pathogen interactions but there is greatest regulation when the fraction born resistant is density independent. Nonetheless, inclusion of density dependence can still allow intrinsically unstable host-pathogen dynamics to be stabilised provided that this effect is sufficiently small. Moreover, inclusion of density-dependent resistance to disease allows the system to give rise to bistable dynamics in which the final outcome is dictated by the initial conditions for the model system. This has implications for the management of agricultural pests using biocontrol agents-in particular, it is suggested that the propensity for density-dependent resistance be determined prior to such a biocontrol attempt in order to be sure that this will result in the prevention of pest outbreaks, rather than their facilitation. Finally we consider how the cost of resistance to disease affects model outcomes and discover that when there is no cost to resistance, the model predicts stable periodic outbreaks of the insect population. The results are interpreted ecologically and future avenues for research to address the shortfalls in the present model system are discussed.     

A deterministic size-structured population model for the worm Naidis elinguis

In this paper we model the population dynamics of the worm Nais elinguis, which reproduces by division into two unequal parts. By using renewal theory we derive the asymptotic behaviour of a Naidis elinguis population. In particular we prove a certain relation between the fraction of the population that was born small (respectively the fraction that was born large) and the inter-division times. Received 20 January 1999 / Revised version: 1 August 1999?Published online: 10 April 2001