The backward bifurcation in compartmental models for West Nile virus

In all of the West Nile virus (WNV) compartmental models in the literature, the basic reproduction number serves as a crucial control threshold for the eradication of the virus. However, our study suggests that backward bifurcation is a common property shared by the available compartmental models with a logistic type of growth for the population of host birds. There exists a subthreshold condition for the outbreak of the virus due to the existence of backward bifurcation. In this paper, we first review and give a comparison study of the four available compartmental models for the virus, and focus on the analysis of the model proposed by Cruz-Pacheco et al. to explore the backward bifurcation in the model. Our comparison study suggests that the mosquito population dynamics itself cannot explain the occurrence of the backward bifurcation, it is the higher mortality rate of the avian host due to the infection that determines the existence of backward bifurcation.     

Diploidy and the selective advantage for sexual reproduction in unicellular organisms

This article develops mathematical models describing the evolutionary dynamics of both asexually and sexually reproducing populations of diploid unicellular organisms. The asexual and sexual life cycles are based on the asexual and sexual life cycles in Saccharomyces cerevisiae, Baker’s yeast, which normally reproduces by asexual budding, but switches to sexual reproduction when stressed. The mathematical models consider three reproduction pathways: (1) Asexual reproduction, (2) self-fertilization, and (3) sexual reproduction. We also consider two forms of genome organization. In the first case, we assume that the genome consists of two multi-gene chromosomes, whereas in the second case, we consider the opposite extreme and assume that each gene defines a separate chromosome, which we call the multi-chromosome genome. These two cases are considered to explore the role that recombination has on the mutation-selection balance and the selective advantage of the various reproduction strategies. We assume that the purpose of diploidy is to provide redundancy, so that damage to a gene may be repaired using the other, presumably undamaged copy (a process known as homologous recombination repair). As a result, we assume that the fitness of the organism only depends on the number of homologous gene pairs that contain at least one functional copy of a given gene. If the organism has at least one functional copy of every gene in the genome, we assume a fitness of 1. In general, if the organism has l homologous pairs that lack a functional copy of the given gene, then the fitness of the organism is κ _l . The κ _l are assumed to be monotonically decreasing, so that κ₀ = 1 > κ₁ > κ₂ > ⋯ > κ_∞ = 0. For nearly all of the reproduction strategies we consider, we find, in the limit of large N, that the mean fitness at mutation-selection balance is max{2 e^-m-1, 0} ,\hbox{max}\{2 e^{-\mu}-1, 0\} , where N is the number of genes in the haploid set of the genome, ε is the probability that a given DNA template strand of a given gene produces a mutated daughter during replication, and μ = Nε. The only exception is the sexual reproduction pathway for the multi-chromosomed genome. Assuming a multiplicative fitness landscape where κ _l  = α ^l for α ∈ (0, 1), this strategy is found to have a mean fitness that exceeds the mean fitness of all the other strategies. Furthermore, while other reproduction strategies experience a total loss of viability due to the steady accumulation of deleterious mutations once μ exceeds ln2 ,\ln 2 , no such transition occurs in the sexual pathway. Indeed, in the limit as α → 1 for the multiplicative landscape, we can show that the mean fitness for the sexual pathway with the multi-chromosomed genome converges to e ^−2μ, which is always positive. We explicitly allow for mitotic recombination in this study, which, in contrast to previous studies using different models, does not have any advantage over other asexual reproduction strategies. The results of this article provide a basis for understanding the selective advantage of the specific meiotic pathway that is employed by sexually reproducing organisms. The results of this article also suggest an explanation for why unicellular organisms such as Saccharomyces cerevisiae (Baker’s yeast) switch to a sexual mode of reproduction when stressed. While the results of this article are based on modeling mutation-propagation in unicellular organisms, they nevertheless suggest that, in more complex organisms with significantly larger genomes, sex is necessary to prevent the loss of viability of a population due to genetic drift. Finally, and perhaps most importantly, the results of this article demonstrate a selective advantage for sexual reproduction with fewer and much less restrictive assumptions than those of previous studies.     

Backward bifurcation,equilibrium and stability phenomena in a three-stage extended BRSV epidemic model

In this paper we consider the phenomenon of backward bifurcation in epidemic modelling illustrated by an extended model for Bovine Respiratory Syncytial Virus (BRSV) amongst cattle. In its simplest form, backward bifurcation in epidemic models usually implies the existence of two subcritical endemic equilibria for R ₀ < 1, where R ₀ is the basic reproductive number, and a unique supercritical endemic equilibrium for R ₀ > 1. In our three-stage extended model we find that more complex bifurcation diagrams are possible. The paper starts with a review of some of the previous work on backward bifurcation then describes our three-stage model. We give equilibrium and stability results, and also provide some biological motivation for the model being studied. It is shown that backward bifurcation can occur in the three-stage model for small b, where b is the common per capita birth and death rate. We are able to classify the possible bifurcation diagrams. Some realistic numerical examples are discussed at the end of the paper, both for b small and for larger values of b.      

Improved methods to detect GTP-binding proteins from plants

Improved methods are described for the detection of G1P-binding proteins (G-proteins) in the protonema of mossFunaria hygrometrica and coleoptiles of corn(Zea mays) and sorghum(Sorghum vulgare). We optimized conditions for the transfer of proteins to nitrocellulose, production of high titer polyclonal anti-Gα (common) antibodies and finally the detection of G-proteins by amplification. In addition to the α-subunit of heterotrimeric G-proteins (M _r 41–43 kDa), a small molecular weight class (< 30 kDa) was also detected by anti-Gα (common) antibodies. An easy, reliable and efficient filter assay is also described to quantify the toxin catalyzed ADP-ribosylation. The apparentK _m of the NAD has been determined to be approximately 1.5μM for the microsomal fraction of moss. Inclusion of G1P stimulated ADP-ribosylation by 2–27-fold. One to three polypeptides representing the α-subunit of heterotrimeric G-proteins of (M_r 37–43 kDa) were ADP-ribosylated in all three plants. The anti-Gβ (C-terminus) antibody cross-reacted strongly with 39 and 34 kDa polypeptide in moss and corn respectively. By employing improved methods two classes of G-proteins have been shown to be present in three plant species.     

Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth

Chronic hepatitis B virus (HBV) infection is a major cause of human suffering, and a number of mathematical models have examined within-host dynamics of the disease. Most previous HBV infection models have assumed that: (a) hepatocytes regenerate at a constant rate from a source outside the liver; and/or (b) the infection takes place via a mass action process. Assumption (a) contradicts experimental data showing that healthy hepatocytes proliferate at a rate that depends on current liver size relative to some equilibrium mass, while assumption (b) produces a problematic basic reproduction number. Here we replace the constant infusion of healthy hepatocytes with a logistic growth term and the mass action infection term by a standard incidence function; these modifications enrich the dynamics of a well-studied model of HBV pathogenesis. In particular, in addition to disease free and endemic steady states, the system also allows a stable periodic orbit and a steady state at the origin. Since the system is not differentiable at the origin, we use a ratio-dependent transformation to show that there is a region in parameter space where the origin is globally stable. When the basic reproduction number, R ₀, is less than 1, the disease free steady state is stable. When R ₀ > 1 the system can either converge to the chronic steady state, experience sustained oscillations, or approach the origin. We characterize parameter regions for all three situations, identify a Hopf and a homoclinic bifurcation point, and show how they depend on the basic reproduction number and the intrinsic growth rate of hepatocytes.     

Effects of excess selenomethionine on selenium status indicators in pregnant long-tailed macaques (Macaca fascicularis)

Forty pregnant long-tailed macaques were treated daily for 30 d with 0, 25, 150 or 300 μg selenium as L-selenomethionine/kg body weight. Erythrocyte and plasma selenium and glutathione peroxidase specific activities, hair and fecal selenium, and urinary selenium excretion were increased by and were linearly related to L-selenomethionine dose. Hair selenium was most sensitive to L-selenomethionine dose, with an 84-fold increase in the 300 μg selenium/(kg-d) group relative to controls (r=0.917). Daily urinary selenium excretion (80-fold,r=0.958), plasma selenium (22-fold,r=0.885), erythrocyte selenium (24-fold,r=0.920), and fecal selenium (18-fold,r=0.911) also responded strongly to L-selenomethionine. Erythrocyte and plasma glutathione peroxidase specific activities increased 154% and 69% over controls, respectively. Toxicity was associated with erythrocyte selenium >2.3 μg/mL, plasma selenium >2.8 μg/mL, and hair selenium >27 μg/g. Plasma, erythrocyte, and hair selenium concentrations may be useful for monitoring and preventing the toxicity of L-selenomethionine administered to humans in cancer chemoprevention trials.     

Joint effects of mitosis and intracellular delay on viral dynamics: two-parameter bifurcation analysis

To understand joint effects of logistic growth in target cells and intracellular delay on viral dynamics in vivo, we carry out two-parameter bifurcation analysis of an in-host model that describes infections of many viruses including HIV-I, HBV and HTLV-I. The bifurcation parameters are the mitosis rate r of the target cells and an intracellular delay τ in the incidence of viral infection. We describe the stability region of the chronic-infection equilibrium E* in the two-dimensional (r, τ) parameter space, as well as the global Hopf bifurcation curves as each of τ and r varies. Our analysis shows that, while both τ and r can destabilize E* and cause Hopf bifurcations, they do behave differently. The intracellular delay τ can cause Hopf bifurcations only when r is positive and sufficiently large, while r can cause Hopf bifurcations even when τ = 0. Intracellular delay τ can cause stability switches in E* while r does not.     

Success rate of a biological invasion in terms of the spatial distribution of the founding population

We analyze the role of the spatial distribution of the initial condition in reaction–diffusion models of biological invasion. Our study shows that, in the presence of an Allee effect, the precise shape of the initial (or founding) population is of critical importance for successful invasion. Results are provided for one-dimensional and two-dimensional models. In the one-dimensional case, we consider initial conditions supported by two disjoint intervals of length L/2 and separated by a distance α. Analytical as well as numerical results indicate that the critical size L ^∗(α) of the population, where the invasion is successful if and only if L>L ^∗(α), is a continuous function of α and tends to increase with α, at least when α is not too small. This result emphasizes the detrimental effect of fragmentation. In the two-dimensional case, we consider more general, stochastically generated initial conditions u ₀, and we provide a new and rigorous definition of the rate of fragmentation of u ₀. We then conduct a statistical analysis of the probability of successful invasion depending on the size of the support of u ₀ and the fragmentation rate of u ₀. Our results show that the outcome of an invasion is almost completely determined by these two parameters. Moreover, we observe that the minimum abundance required for successful invasion tends to increase in a non-linear fashion with the fragmentation rate. This effect of fragmentation is enhanced as the strength of the Allee effect is increased.     

Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations

In nonlinear matrix models, strong Allee effects typically arise when the fundamental bifurcation of positive equilibria from the extinction equilibrium at r=1 (or R₀=1) is backward. This occurs when positive feedback (component Allee) effects are dominant at low densities and negative feedback effects are dominant at high densities. This scenario allows population survival when r (or equivalently R₀) is less than 1, provided population densities are sufficiently high. For r>1 (or equivalently R₀>1) the extinction equilibrium is unstable and a strong Allee effect cannot occur. We give criteria sufficient for a strong Allee effect to occur in a general nonlinear matrix model. A juvenile–adult example model illustrates the criteria as well as some other possible phenomena concerning strong Allee effects (such as positive cycles instead of equilibria).     

Progression age enhanced backward bifurcation in an epidemic model with super-infection

 We consider a model for a disease with a progressing and a quiescent exposed class and variable susceptibility to super-infection. The model exhibits backward bifurcations under certain conditions, which allow for both stable and unstable endemic states when the basic reproduction number is smaller than one. Received: 11 October 2001 / Revised version: 17 September 2002 / Published online: 17 January 2003 Present address: Department of Biological Statistics and Computational Biology, 434 Warren Hall, Cornell University, Ithaca, NY 14853-7801 This author was visiting Arizona State University when most of the research was done. Research partially supported by NSF grant DMS-0137687. This author's research was partially supported by NSF grant DMS-9706787. Key words or phrases: Backward bifurcation – Multiple endemic equilibria – Alternating stability – Break-point density – Super-infection – Dose-dependent latent period – Progressive and quiescent latent stages – Progression age structure – Threshold type disease activation – Operator semigroups – Hille-Yosida operators – Dynamical systems – Persistence – Global compact attractor     

A metapopulation model for malaria with transmission-blocking partial immunity in hosts

A metapopulation malaria model is proposed using SI and SIRS models for the vectors and hosts, respectively. Recovered hosts are partially immune to the disease and while they cannot directly become infectious again, they can still transmit the parasite to vectors. The basic reproduction number R₀{\mathcal{R}_0} is shown to govern the local stability of the disease free equilibrium but not the global behavior of the system because of the potential occurrence of a backward bifurcation. Using type reproduction numbers, we identify the reservoirs of infection and evaluate the effect of control measures. Applications to the spread to non-endemic areas and the interaction between rural and urban areas are given.     

Seminal plasma zinc concentration and α-glucosidase activity with respect to semen quality

This study was conducted to examine the relationship between the concentration of zinc and neutral α-glucosidase (NAG) with semen quality. Semen samples from 75 male partners of couples who were attending the Obstetrics and Gynecology Department were analyzed for semen quality. Based on sperm count, the subjects were divided into three groups. Zinc and neutral α-glucosidase activity were estimated in seminal plasma. Results showed that mean the α-glucosidase activity was lowest among the azoospermic group with respect to oligozoospermic and normozoospermic groups. Mean zinc levels were also lower among azoospermics compared to oligozoospermic and normospermic groups. The difference was statistically significant (p<0.05). A significant positive correlation was observed between zinc levels and sperm count (r=0.29, p<0.05) and zinc and α-glucosidase activity (r=0.31, p<0.05) in seminal plasma. These results suggest that zinc and neutral α-glucosidase seem to play an important role in human reproduction.     

Analysis of 3D structural differences in the IgG-binding domains based on the interresidue average-distance statistics

It is well-known that the IgG-binding domain from staphylococcal protein A folds into a 3α helix bundle structure, while the IgG-binding domain of streptococcal protein G forms an (α + β) structure. Recently, He et al. (Biochemistry 44:14055–14061, 2005) made mutants of these proteins from the wild types of protein A and protein G strains. These mutants are referred to as protein A219 and protein G311, and it was showed that these two mutants have different 3D structures, i.e., the 3α helix bundle structure and the (α + β) structure, respectively, despite the high sequence identity (59%). The purpose of our study was to clarify how such 3D structural differences are coded in the sequences with high homology. To address this problem, we introduce a predicted contact map constructed based on the interresidue average-distance statistics for prediction of folding properties of a protein. We refer to this map as an average distance map (ADM). Furthermore, the statistics of interresidue distances can be converted to an effective interresidue potential. We calculated the contact frequency of each residue of a protein in random conformations with this effective interresidue potential, and then we obtained values similar to ϕ values. We refer to this contact frequency of each residue as a p(μ) value. The comparison of the p(μ) values to the ϕ values for a protein suggests that p(μ) values reveal the information on the folding initiation site. Using these techniques, we try to extract the information on the difference in the 3D structures of protein A219 and protein G311 coded in their amino acid sequences in the present work. The results show that the ADM analyses and the p(μ) value analyses predict the information of folding initiation sites, which can be used to detect the 3D difference in both proteins.     

Cervical mucins carry α(1,2)fucosylated glycans that partly protect from experimental vaginal candidiasis

Cervical mucins are glycosylated proteins that form a protective cervical mucus. To understand the role of mucin glycans in Candida albicans infection, oligosaccharides from mouse cervical mucins were analyzed by liquid chromatography-mass spectrometry. Cervical mucins carry multiple α(1-2)fucosylated glycans, but α(1,2)fucosyltransferase Fut2-null mice are devoid of these epitopes. Epithelial cells in vaginal lavages from Fut2-null mice lacked Ulex europaeus agglutinin-1 (UEA-I) staining for α(1-2)fucosylated glycans. Hysterectomy to remove cervical mucus eliminated UEA-I and acid mucin staining in vaginal epithelial cells from wild type mice indicating the cervix as the source of UEA-I positive epithelial cells. To assess binding of α(1-2) fucosylated glycans on C. albicans infection, an in vitro adhesion assay was performed with vaginal epithelial cells from wild type and Fut2-null mice. Vaginal epithelial cells from Fut2-null mice were found to bind increased numbers of C. albicans compared to vaginal epithelial cells obtained from wild type mice. Hysterectomy lessened the difference between Fut2-null and wild type mice in binding of C. ablicans in vitro and susceptibility to experimental C. albicans vaginitis in vivo. We generated a recombinant fucosylated MUC1 glycanpolymer to test whether the relative protection of wild type mice compared to Fut2-null mice could be mimicked with exogenous mucin. While a small portion of the recombinant MUC1 epitopes displayed α(1-2)fucosylated glycans, the predominant epitopes were sialylated due to endogenous sialyltransferases in the cultured cells. Intravaginal instillation of recombinant MUC1 glycanpolymer partially reduced experimental yeast vaginitis suggesting that a large glycanpolymer, with different glycan epitopes, may affect fungal burden.     

Modeling and dynamics of Wolbachia-infected male releases and mating competition on mosquito control

Despite centuries of continuous efforts, mosquito-borne diseases (MBDs) remain enormous health threat of human life worldwide. Lately, the USA government has approved an innovative technology of releasing Wolbachia-infected male mosquitoes to suppress the wild mosquito population. In this paper we first introduce a stage-structured model for natural mosquitos, then we establish a new model considering the releasing of Wolbachia-infected male mosquitoes and the mating competition between the natural male mosquitoes and infected males on the suppression of natural mosquitoes. Dynamical analysis of the two models, including the existence and local stability of the equilibria and bifurcation analysis, reveals the existence of a forward bifurcation or a backward bifurcation with multiple attractors. Moreover, globally dynamical properties are further explored by using Lyapunov function and theory of monotone operators, respectively. Our findings suggest that infected male augmentation itself cannot always guarantee the success of population eradication, but leads to three possible levels of population suppression, so we define the corresponding suppression rate and estimate the minimum release ratio for population eradication. Furthermore, we study how the release ratio of infected males and natural ones, mating competition, the rate of cytoplasmic incompatibility and the basic offspring number affect the suppression rate of natural mosquitoes. Our results show that the successful eradication relies on assessing the reproductive capacity of natural mosquitoes, a selection of suitable Wolbachia strains and an appropriate release amount of infected males. This study will be helpful for public health authorities in designing proper strategies to control vector mosquitoes and prevent the epidemics of MBDs.

     

Dimensionless analysis of the microbial growth rate dependence on sub-optimal temperatures

The influence of sub-optimal temperatures (T) on the microbial growth rate (μ) has been assessed by means of dimensionless variables: T_dim = [T−T_min]/[T_opt−T_min] and μ_dim = μ/μ_opt. T_min represents the temperature at which there is no growth, T_opt the optimum temperature at which the growth rate, μ_opt, is maximum. Data sets, growth rate vs temperature, have been taken from the literature for 12 organisms (psychrotrophs, mesophiles and thermophiles). In order to compare these organisms, the power law function has been used: [μ_dim] = [T_dim]^α. The parameters μ_opt and T_opt are determined from direct readings whereas T_min and αare estimated by means of a non-linear regression. An accurate estimation of T_min is obtained providing low growth rate data are available. A wide range of optimal temperatures where the growth rate almost equals μ_opt prevents one from obtaining a narrow confidence interval forα. On the basis of the analysis hereafter developed, thermophiles are characterized by values of the power α less than mesophiles and psychrotrophs. Almost all of these values are significantly different from two, previously determined for Staphylococcus xylosus and widely used for predicting the microbial growth in foods. Received 15 May 1998/ Accepted in revised form 25 September 1998     

Critical Levels of Selenium in Different Crops Grown in an Alkaline Silty Loam Soil Treated with Selenite-Se

Critical levels of selenium in raya (Brassica juncea Czern L.), maize (Zea mays L.), wheat (Triticum aestivum L.) and rice (Oryza sativa L.) were worked out by growing these crops in an alkaline silty loam soil treated with different levels of selenite-Se ranging from 1 to 25 μg g⁻¹ soil. Significant decrease in dry matter yield was observed above a level of 5 μg Se g⁻¹ soil in raya and maize; 4 μg Se g⁻¹ soil in wheat and 10 μg Se g⁻¹ soil in rice shoots. The critical level of Se in plants above which significant decrease in yield would occur was found to be 104.8 μg g⁻¹ in raya, 76.9 μg g⁻¹ in maize, 41.5 μg g⁻¹ in rice and 18.9 μg g⁻¹ in wheat shoots. Significant coefficients of correlation were observed between Se content above the critical level and dry matter yield of raya as well as rice (r = −0.99, P ≤ 0.01), wheat (r = −0.97, P ≤ 0.01) and maize ((r = −0.96, P ≤ 0.01). A synergistic relationship was observed between S and Se content of raya (r = 0.96, P ≤ 0.01), wheat (r = 0.89, P ≤ 0.01), rice (r = 0.85, P ≤ 0.01) and maize (r = 0.84, P ≤ 0.01). Raya, maize and rice absorbed Se in levels toxic for animal consumption (i.e. > 5 mg Se kg⁻¹) when the soil was treated with more than 1.5 μg Se g⁻¹. In case of wheat, application of Se more than 3 μg g⁻¹ soil resulted in production of toxic plants.     

Stability of differential susceptibility and infectivity epidemic models

We introduce classes of differential susceptibility and infectivity epidemic models. These models address the problem of flows between the different susceptible, infectious and infected compartments and differential death rates as well. We prove the global stability of the disease free equilibrium when the basic reproduction ratio R₀ ￡ 1{\mathcal{R}_0 \leq 1} and the existence and uniqueness of an endemic equilibrium when ${\mathcal{R}_0 >1 }${\mathcal{R}_0 >1 } . We also prove the global asymptotic stability of the endemic equilibrium for a differential susceptibility and staged progression infectivity model, when ${\mathcal{R}_0 >1 }${\mathcal{R}_0 >1 } . Our results encompass and generalize those of Hyman and Li (J Math Biol 50:626–644, 2005; Math Biosci Eng 3:89–100, 2006).     

Concentration-dependent differntial effects of cortisol on synthesis of α-lactalbumin and of casein in cultured mouse mammary gland explants: Importance of prolactin concentration

Summary Cortisol was previously shown to elicit a concentration-dependent inhibition of α-lactalbumin accumulation in midpregnant mouse mammary gland cultured in medium containing optimal concentrations of 5 μg/ml prolactin and insulin. In contrast, casein accumulation under these conditions was progressively stimulated by addition of increasing amounts of cortisol (Ono, M.; Oka, T. Cell 19: 473–480; 1980). In the present study we found that in the presence of a suboptimal concentration of 0.5 μg/ml prolactin, 2.8×10⁻⁹ M to 2.8×10⁻⁷ M cortisol stimulated α-lactalbumin accumulation. Furthermore, higher concentrations of cortisol produced a smaller inhibition of α-lactalbumin accumulation as compared to that obtained in cultures containing 5 μg/ml prolactin. The maximal increase in α-lactalbumin accumulation attained in the presence of 1.4×10⁻⁸ M cortisol, 0.5 μg/ml prolactin, and insulin was comparable to that observed in culture containing 5 μg/ml prolactin and insulin. Similar results were obtained in a cortisol concentration-response study of α-lactalbumin accumulation in cultures containing a suboptimal concentration of 0.5 μg/ml human placental lactogen. Measurement of the rate of α-lactalbumin synthesis in cultured tissue indicated that the opposing effects of low and high concentrations of cortisol on α-lactalbumin accumulation involved an alteration in the rate of synthesis of the milk protein. In contrast to α-lactalbumin, the synthesis of casein was stimulated in a concentration-dependent manner by addition of cortisol that acted synergistically with either 0.5 μg/ml or 5 μg/ml prolactin. The maximal increases were obtained in the presence of 2.8×10⁻⁶ M cortisol. These results indicated that the action of cortisol on α-lactalbumin accumulation can be modulated by the concentration, of prolactin and suggest that the interplay between cortisol and prolactin in regulation of α-lactalbumin synthesis may be different from that involved in casein synthesis.