Cell Volume Regulation in the Proximal Tubule of Rat Kidney

We developed a dynamic model of a rat proximal convoluted tubule cell in order to investigate cell volume regulation mechanisms in this nephron segment. We examined whether regulatory volume decrease (RVD), which follows exposure to a hyposmotic peritubular solution, can be achieved solely via stimulation of basolateral K\(^+\) and \(\hbox {Cl}^-\) channels and \(\hbox {Na}^+\)–\(\hbox {HCO}_3^-\) cotransporters. We also determined whether regulatory volume increase (RVI), which follows exposure to a hyperosmotic peritubular solution under certain conditions, may be accomplished by activating basolateral \(\hbox {Na}^+\)/H\(^+\) exchangers. Model predictions were in good agreement with experimental observations in mouse proximal tubule cells assuming that a 10% increase in cell volume induces a fourfold increase in the expression of basolateral K\(^+\) and \(\hbox {Cl}^-\) channels and \(\hbox {Na}^+\)–\(\hbox {HCO}_3^-\) cotransporters. Our results also suggest that in response to a hyposmotic challenge and subsequent cell swelling, \(\hbox {Na}^+\)–\(\hbox {HCO}^-_3\) cotransporters are more efficient than basolateral K\(^+\) and \(\hbox {Cl}^-\) channels at lowering intracellular osmolality and reducing cell volume. Moreover, both RVD and RVI are predicted to stabilize net transcellular \(\hbox {Na}^+\) reabsorption, that is, to limit the net \(\hbox {Na}^+\) flux decrease during a hyposmotic challenge or the net \(\hbox {Na}^+\) flux increase during a hyperosmotic challenge.     

Isolation and characterization of microsatellite markers in the spiny lobster, <Emphasis Type="Italic">Panulirus echinatus</Emphasis> Smith, 1869 (Decapoda: Palinuridae) by Illumina MiSeq sequencing

Okra’s (Abelmoschus esculentus (L.) Moench) commercial cultivation is threatened in the tropics due to high incidence of yellow vein mosaic virus (YVMV) disease. Okra geneticists across the world tried to understand the inheritance pattern of YVMV disease tolerance without much success. Therefore, the inheritance pattern of YVMV disease in okra was revisited by employing six generations (\(\hbox {P}_{1}\), \(\hbox {P}_{2}\), \(\hbox {F}_{1}\), \(\hbox {F}_{2}\), \(\hbox {BC}_{1}\) and \(\hbox {BC}_{2}\)) of four selected crosses (one tolerant \(\times \) tolerant, two tolerant \(\times \) susceptible and one susceptible \(\times \) susceptible) using two tolerant (BCO-1 and Lal Bhendi) and two susceptible (Japanese Jhar Bhendi and PAN 2127) genotypes. Qualitative genetic analysis was done on the basis of segregation pattern of tolerant and susceptible plants in \(\hbox {F}_{2}\) and backcross generations of all the four crosses. It revealed that a single dominant gene along with some minor factors governed the disease tolerant trait in both the tolerant parents used. However, it was observed that genes governing disease tolerance identified in both the tolerant variety used was different. It could be concluded that the gene governing YVMV disease tolerance in okra was genotype specific. Further, duplicate gene action as evident from an approximate ratio of 15 : 1 (tolerant : susceptible) in the \(\hbox {F}_{2}\) population in the cross of two tolerant varieties gave a scope of increasing the tolerance level of the hybrid plants when both the tolerant genes are brought together. However, generation mean analysis revealed involvement of both additive and nonadditive effects in the inheritance of disease tolerance. Thus, the present study confirms that a complicated genetic inheritance pattern is involved in the disease tolerance against YVMV trait. The major tolerance genes could be transferred to other okra varieties, but the tolerance breaking virus strains might not allow them to achieve tolerance in stable condition. Therefore, accumulation of additional genes may be needed for a sustainable tolerance phenotype in okra.     

Optimal foraging behavior with an explicit consideration of within-individual behavioral variation: an example of predation

Animal behavior is flexible, and the same individual can exhibit variable expressions under the equivalent ecological situations (i.e., within-individual behavioral variation). This study examines the evolution of within-individual behavioral variation using an individual-based model. A common predation scenario is considered where a predator spends a period h to handle and consume a captured prey. The model assumes the handling time of the predator to be a random variable. The average and within-individual variance of handling time are described by \(\mu _h\) and \(\sigma _h^2\), respectively, where each individual has its own unique \(\mu _h\) and \(\sigma _h^2\). Using a genetic algorithm, the evolution of \(\sigma _h^2\) is traced. The results show that natural selection acts on both \(\mu _h\) and \(\sigma _h^2\), and the optimal behavioral variation depends on the density of prey. In particular, individuals with high behavioral variance \(\sigma _h^2\) are more likely selected when prey density is low. Individual based modeling can be a useful tool for studying the ultimate significance of within-individual behavioral variation and generating empirically testable predictions. The mechanisms of the evolution of within-individual behavioral variation and their ecological implications are discussed.     

A Mathematical Model Supports a Key Role for Ae4 (Slc4a9) in Salivary Gland Secretion

We develop a mathematical model of a salivary gland acinar cell with the objective of investigating the role of two \(\mathrm{Cl}^-/\mathrm{HCO}_3^-\) exchangers from the solute carrier family 4 (Slc4), Ae2 (Slc4a2) and Ae4 (Slc4a9), in fluid secretion. Water transport in this type of cell is predominantly driven by \(\mathrm{Cl}^-\) movement. Here, a basolateral \(\mathrm{Na}^+/ \mathrm{K}^+\) adenosine triphosphatase pump (NaK-ATPase) and a \(\mathrm{Na}^+\)–\(\mathrm{K}^+\)–\(2 \mathrm{Cl}^-\) cotransporter (Nkcc1) are primarily responsible for concentrating the intracellular space with \(\mathrm{Cl}^-\) well above its equilibrium potential. Gustatory and olfactory stimuli induce the release of \(\mathrm{Ca}^{2+}\) ions from the internal stores of acinar cells, which triggers saliva secretion. \(\mathrm{Ca}^{2+}\)-dependent \(\mathrm{Cl}^-\) and \(\mathrm{K}^+\) channels promote ion secretion into the luminal space, thus creating an osmotic gradient that promotes water movement in the secretory direction. The current model for saliva secretion proposes that \(\mathrm{Cl}^-/ \mathrm{HCO}_3^-\) anion exchangers (Ae), coupled with a basolateral \(\mathrm{Na}^+/\hbox {proton}\) (\(\hbox {H}^+\)) (Nhe1) antiporter, regulate intracellular pH and act as a secondary \(\mathrm{Cl}^-\) uptake mechanism (Nauntofte in Am J Physiol Gastrointest Liver Physiol 263(6):G823–G837, 1992; Melvin et al. in Annu Rev Physiol 67:445–469, 2005.  https://doi.org/10.1146/annurev.physiol.67.041703.084745). Recent studies demonstrated that Ae4 deficient mice exhibit an approximate \(30\%\) decrease in gland salivation (Peña-Münzenmayer et al. in J Biol Chem 290(17):10677–10688, 2015). Surprisingly, the same study revealed that absence of Ae2 does not impair salivation, as previously suggested. These results seem to indicate that the Ae4 may be responsible for the majority of the secondary \(\mathrm{Cl}^-\) uptake and thus a key mechanism for saliva secretion. Here, by using ‘in-silico’ Ae2 and Ae4 knockout simulations, we produced mathematical support for such controversial findings. Our results suggest that the exchanger’s cotransport of monovalent cations is likely to be important in establishing the osmotic gradient necessary for optimal transepithelial fluid movement.     

Backbone resonance assignments for the SET domain of the human methyltransferase NSD2

Aberrant NSD2 methyltransferase activity is implicated as the oncogenic driver in multiple myeloma, suggesting opportunities for novel therapeutic intervention. The methyltransferase activity of NSD2 resides in its catalytic SET domain, which is conserved among most lysine methyltransferases. Here we report the backbone \(\hbox {H}^{\mathrm{N}}\), N, C\(^{\prime }\), \(\hbox {C}^\alpha\) and side-chain \(\hbox {C}^\beta\) assignments of a 25 kDa NSD2 SET domain construct, spanning residues 991–1203. A chemical shift analysis of C\(^{\prime }\), \(\hbox {C}^\alpha\) and \(\hbox {C}^\beta\) resonances predicts a secondary structural pattern that is in agreement with homology models.     

Analysis of two susceptibility SNPs in HLA region and evidence of interaction between rs6457617 in <Emphasis Type="BoldItalic">HLA-DQB1</Emphasis> and <Emphasis Type="BoldItalic">HLA-DRB1</Emphasis>*04 locus on Tunisian rheumatoid arthritis

Previous genomewide association studies (GWAS) and meta-analyses have enumerated several genes/loci in major histocompatibility complex region, which are consistently associated with rheumatoid arthritis (RA) in different ethnic populations. Given the genetic heterogeneity of the disease, it is necessary to replicate these susceptibility loci in other populations. In this case, we investigate the analysis of two SNPs, rs13192471 and rs6457617, from the human leukocyte antigen (HLA) region with the risk of RA in Tunisian population. These SNPs were previously identified to have a strong RA association signal in several GWAS studies. A case–control sample composed of 142 RA patients and 123 healthy controls was analysed. Genotyping of rs13192471 and rs6457617 was carried out using real-time PCR methods by TaqMan allelic discrimination assay. A trend of significant association was found in rs6457617 TT genotype with susceptibility to RA (\(P = 0.04\), \(p_{c} = 0.08\), \(\hbox {OR} = 1.73\)). Moreover, using multivariable analysis, the combination of rs6457617*TT–HLA-DRB1*\(04^{+}\) increased risk of RA (\(\hbox {OR} = 2.38\)), which suggest a gene–gene interaction event between rs6457617 located within the HLA-DQB1 and HLA-DRB1. Additionally, haplotypic analysis highlighted a significant association of rs6457617*T–HLA-DRB1*\(04^{+}\) haplotype with susceptibility to RA (\(P = 0.018\), \(p_{c} = 0.036\), \(\hbox {OR} = 1.72\)). An evidence of association was shown subsequently in \(\hbox {antiCCP}^{+}\) subgroup with rs6457617 both in T allele and TT genotype (\(P = 0.01\), \(p_{c} = 0.03\), \(\hbox {OR} = 1.66\) and \(P = 0.008\), \(p_{c} = 0.024\), \(\hbox {OR} = 1.28\), respectively). However, no association was shown for rs13192471 polymorphism with susceptibility and severity to RA. This study suggests the involvement of rs6457617 locus as risk variant for susceptibility/severity to RA in Tunisian population. Secondly, it highlights the gene–gene interaction between HLA-DQB1 and HLA-DRB1.     

A novel real-time driving fatigue detection system based on wireless dry EEG

Development of techniques for detection of mental fatigue has varied applications in areas where sustaining attention is of critical importance like security and transportation. The objective of this study is to develop a novel real-time driving fatigue detection methodology based on dry Electroencephalographic (EEG) signals. The study has employed two methods in the online detection of mental fatigue: power spectrum density (PSD) and sample entropy (SE). The wavelet packets transform (WPT) method was utilized to obtain the \(\theta \) (4–7 Hz), \(\alpha \) (8–12 Hz) and \(\beta \) (13–30 Hz) bands frequency components for calculating corresponding PSD of the selected channels. In order to improve the fatigue detection performance, the system was individually calibrated for each subject in terms of fatigue-sensitive channels selection. Two fatigue-related indexes: (\(\theta +\alpha \))/\(\beta \) and \(\theta \)/\(\beta \) were computed and then fused into an integrated metric to predict the degree of driving fatigue. In the case of SE extraction, the mean of SE averaged across two EEG channels (‘O1h’ and ‘O2h’) was used for fatigue detection. Ten healthy subjects participated in our study and each of them performed two sessions of simulated driving. In each session, subjects were required to drive simulated car for 90 min without any break. The results demonstrate that our proposed methods are effective for fatigue detection. The prediction of fatigue is consistent with the observation of reaction time that was recorded during simulated driving, which is considered as an objective behavioral measure.     

Cellular mechanosensitivity to substrate stiffness decreases with increasing dissimilarity to cell stiffness

Computational modelling has received increasing attention to investigate multi-scale coupled problems in micro-heterogeneous biological structures such as cells. In the current study, we investigated for a single cell the effects of (1) different cell-substrate attachment (2) and different substrate modulus \(\textit{E}_\mathrm{s}\) on intracellular deformations. A fibroblast was geometrically reconstructed from confocal micrographs. Finite element models of the cell on a planar substrate were developed. Intracellular deformations due to substrate stretch of \(\lambda =1.1\), were assessed for: (1) cell-substrate attachment implemented as full basal contact (FC) and 124 focal adhesions (FA), respectively, and \(\textit{E}_\mathrm{s}\,=\,\)140 KPa and (2) \(\textit{E}_\mathrm{s}\,=\,10\), 140, 1000, and 10,000 KPa, respectively, and FA attachment. The largest strains in cytosol, nucleus and cell membrane were higher for FC (1.35\(\text {e}^{-2}\), 0.235\(\text {e}^{-2}\) and 0.6\(\text {e}^{-2}\)) than for FA attachment (0.0952\(\text {e}^{-2}\), 0.0472\(\text {e}^{-2}\) and 0.05\(\text {e}^{-2}\)). For increasing \(\textit{E}_\mathrm{s}\), the largest maximum principal strain was 4.4\(\text {e}^{-4}\), 5\(\text {e}^{-4}\), 5.3\(\text {e}^{-4}\) and 5.3\(\text {e}^{-4}\) in the membrane, 9.5\(\text {e}^{-4}\), 1.1\(\text {e}^{-4}\), 1.2\(\text {e}^{-3}\) and 1.2\(\text {e}^{-3}\) in the cytosol, and 4.5\(\text {e}^{-4}\), 5.3\(\text {e}^{-4}\), 5.7\(\text {e}^{-4}\) and 5.7\(\text {e}^{-4}\) in the nucleus. The results show (1) the importance of representing FA in cell models and (2) higher cellular mechanical sensitivity for substrate stiffness changes in the range of cell stiffness. The latter indicates that matching substrate stiffness to cell stiffness, and moderate variation of the former is very effective for controlled variation of cell deformation. The developed methodology is useful for parametric studies on cellular mechanics to obtain quantitative data of subcellular strains and stresses that cannot easily be measured experimentally.     

Genomewide identification of PPR gene family and prediction analysis on restorer gene in <Emphasis Type="BoldItalic">Gossypium</Emphasis>

Pentatricopeptide repeat (PPR) gene family plays an essential role in the regulation of plant growth and organelle gene expression. Some PPR genes are related to fertility restoration in plant, but there is no detailed information in Gossypium. In the present study, we identified 482 and 433 PPR homologues in Gossypium raimondii (\(\hbox {D}_{5}\)) and G. arboreum (\(\hbox {A}_{2}\)) genomes, respectively. Most PPR homologues showed an even distribution on the whole chromosomes. Given an evolutionary analysis to PPR genes from G. raimondii (\(\hbox {D}_{5}\)), G. arboreum (\(\hbox {A}_{2}\)) and G. hirsutum genomes, eight PPR genes were clustered together with restoring genes of other species. Most cotton PPR genes were qualified with no intron, high proportion of \(\upalpha \)-helix and classical tertiary structure of PPR protein. Based on bioinformatics analyses, eight PPR genes were targeted in mitochondrion, encoding typical P subfamily protein with protein binding activity and organelle RNA metabolism in function. Further verified by RNA-seq and quantitative real-time PCR (qRT-PCR) analyses, two PPR candidate genes, Gorai.005G0470 (\(\hbox {D}_{5}\)) and Cotton_A_08373 (\(\hbox {A}_{2}\)), were upregulated in fertile line than sterile line. These results reveal new insights into PPR gene evolution in Gossypium.     

Glutathione <Emphasis Type="BoldItalic">S</Emphasis>-transferase P1 gene polymorphisms and susceptibility to coronary artery disease in a subgroup of north Indian population

The present study aimed to investigate the association of \(\hbox {g}.313\hbox {A}{>}\hbox {G}\) and \(\hbox {g}.341\hbox {C}{>}\hbox {T}\) polymorphisms of GSTP1 with coronary artery disease (CAD) in a subgroup of north Indian population. In the present case–control study, CAD patients (\(n = 200\)) and age-matched, sex-matched and ethnicity-matched healthy controls (\(n = 200\)) were genotyped for polymorphisms in GSTP1 using polymerase chain reaction-restriction fragment length polymorphism (PCR-RFLP) method. Genotype distribution of \(\hbox {g}.313\hbox {A}{>}\hbox {G}\) and \(\hbox {g}.341\hbox {C}{>}\hbox {T}\) polymorphisms of GSTP1 gene was significantly different between cases and controls (\(P = 0.005\) and 0.024, respectively). Binary logistic regression analysis showed significant association of A/G (odds ratio (OR): 1.6, 95% CI: 1.08–2.49, \(P = 0.020\)) and G/G (OR: 3.1, 95% CI: 1.41–6.71, P \(=\) 0.005) genotypes of GSTP1 \(\hbox {g}.313\hbox {A}{\!>\!}\hbox {G}\), and C/T (OR: 5.8, 95% CI: 1.26–26.34, \(P = 0.024\)) genotype of GSTP1 \(\hbox {g}.341\hbox {C}{>}\hbox {T}\) with CAD. The A/G and G/G genotypes of \(\hbox {g}.313\hbox {A}{>}\hbox {G}\) and C/T genotype of \(\hbox {g}.341\hbox {C}{>}\hbox {T}\) conferred 6.5-fold increased risk for CAD (OR: 6.5, 95% CI: 1.37–31.27, \(P = 0.018\)). Moreover, the recessive model of GSTP1 \(\hbox {g}.313\hbox {A}{>}\hbox {G}\) is the best fit inheritance model to predict the susceptible gene effect (OR: 2.3, 95% CI: 1.11–4.92, \(P = 0.020\)). In conclusion, statistically significant associations of GSTP1 \(\hbox {g}.313\hbox {A}{>}\hbox {G}\) (A/G, G/G) and \(\hbox {g}.341\hbox {C}{>}\hbox {T}\) (C/T) genotypes with CAD were observed.     

A finite-element model of mechanosensation by a Pacinian corpuscle cluster in human skin

The Pacinian corpuscle (PC) is the cutaneous mechanoreceptor responsible for sensation of high-frequency (20–1000 Hz) vibrations. PCs lie deep within the skin, often in multicorpuscle clusters with overlapping receptive fields. We developed a finite-element mechanical model of one or two PCs embedded within human skin, coupled to a multiphysics PC model to simulate action potentials elicited by each PC. A vibration was applied to the skin surface, and the resulting mechanical signal was analyzed using two metrics: the deformation amplitude ratio (\({\rho }_{\mathrm{1S}} \), \({\rho }_{\mathrm{2S}} )\) and the phase shift of the vibration (\({\delta }_{\mathrm{S}1}^{\mathrm{mech}} \), \({\delta }_{\mathrm{S}2}^{\mathrm{mech}} )\) between the stimulus and the PC. Our results showed that the amplitude attenuation and phase shift at a PC increased with distance from the stimulus to the PC. Differences in amplitude (\(\rho _{12} )\) and phase shift (\({\delta }_{12}^{\mathrm{mech}} )\) between the two PCs in simulated clusters directly affected the interspike interval between the action potentials elicited by each PC (\({\delta }_{12}^{\mathrm{spike}} )\). While \({\delta }_{12}^{\mathrm{mech}} \) had a linear relationship with \({\delta }_{12}^{\mathrm{spike}} \), \(\rho _{12} \)’s effect on \({\delta }_{12}^{\mathrm{spike}} \) was greater for lower values of \(\rho _{12} \). In our simulations, the separation between PCs and the distance of each PC from the stimulus location resulted in differences in amplitude and phase shift at each PC that caused \({\delta }_{12}^{\mathrm{spike}} \) to vary with PC location. Our results suggest that PCs within a cluster receive different mechanical stimuli which may enhance source localization of vibrotactile stimuli, drawing parallels to sound localization in binaural hearing.     

Trend analysis of variations in carbon stock using stock big data

Changes in land use affect the terrestrial carbon stock through changes in the land cover. Research on land use and analysis of variations in carbon stock have practical applications in the optimization of land use and the mitigation of climate change effects. This study was conducted in Baixiang and Julu counties in the Taihang Piedmont by employing the trend analysis method to characterize the variation in county land use and carbon stock. The findings show that in both counties, agricultural and unused land areas decreased while built-up land area increased, and the reduction in cropland was the main reason behind the agricultural land reduction. An inflection point appeared on the cropland curves of Julu, because the cropland area decreased by 1576.97 hm\(^{2}\) from 2004 to 2006. Cropland area in Baixiang decreased from 1996 to 1998 by a total of 129.89 hm\(^{2}\) and then remained relatively stable after 1998. The total carbon storage and variation in land use in the two counties displayed similar trends. Total carbon reserves in Julu increased by 2.76 \(\times \) 10\(^{4}\) tC (carbon equivalent), while those in Baixiang decreased by 0.63 \(\times \) 10\(^{4}\) tC. Carbon stock of built-up land in Julu and Baixiang increased by 2.44 \(\times \) 10\(^{4}\) and 1.22 \(\times \) 10\(^{4}\) tC, respectively.     

A Mathematical Model for the Macrophage Response to Respiratory Viral Infection in Normal and Asthmatic Conditions

Respiratory viral infections are common in the general population and one of the most important causes of asthma aggravation and exacerbation. Despite many studies, it is not well understood how viral infections cause more severe symptoms and exacerbations in asthmatics. We develop a mathematical model of two types of macrophages that play complementary roles in fighting viral infection: classically \((\hbox {CA}\)-\(\hbox {M}\Phi )\) and alternatively activated macrophages \((\hbox {AA}\)-\(\hbox {M}\Phi )\). \(\hbox {CA}\)-\(\hbox {M}\Phi \) destroy infected cells and tissues to remove viruses, while \(\hbox {AA}\)-\(\hbox {M}\Phi \) repair damaged tissues. We show that a higher viral load or longer duration of infection provokes a stronger immune response from the macrophage system. By adjusting the parameters, we model the differences in response to respiratory viral infection in normal and asthmatic subjects and show how this skews the system toward a response that generates more severe symptoms in asthmatic patients.     

Tied up: Does altering phosphoinositide-mediated membrane trafficking influence neurodegenerative disease phenotypes?

Phosphoinositides are a class of membrane lipids that are found on several intracellular compartments and play diverse roles inside cells, such as vesicle formation, protein trafficking, endocytosis etc. Intracellular distribution and levels of phosphoinositides are regulated by enzymes that generate and breakdown these lipids as well as other proteins that associate with phosphoinositides. These events lead to differing levels of specific phosphoinositides on different intracellular compartments. At these intracellular locations, phosphoinositides and their associated proteins, such as Rab GTPases, dynamin and BAR domain-containing proteins, regulate a variety of membrane trafficking pathways. Neurodegenerative phenotypes in disorders such as Parkinson’s disease (PD) can arise as a consequence of altered or hampered intracellular trafficking. Altered trafficking can cause proteins such as \(\upalpha \)-synuclein to aggregate intracellularly. Several trafficking pathways are regulated by master regulators such as LRRK2, which is known to regulate the activity of phosphoinositide effector proteins. Perturbing either the levels of phosphoinositides or their interactions with different proteins disrupts intracellular trafficking pathways, contributing to phenotypes often observed in disorders such as Alzheimer’s or PDs. Thus, studying phosphoinositide regulation and its role in trafficking can give us a deeper understanding of the contribution of disrupted trafficking to neurodegenerative phenotypes.     

Influence of muscle groups’ activation on proximal femoral growth tendency

Muscle and joint contact force influence stresses at the proximal growth plate of the femur and thus bone growth, affecting the neck shaft angle (NSA) and femoral anteversion (FA). This study aims to illustrate how different muscle groups’ activation during gait affects NSA and FA development in able-bodied children. Subject-specific femur models were developed for three able-bodied children (ages 6, 7, and 11 years) using magnetic resonance images. Contributions of different muscle groups—hip flexors, hip extensors, hip adductors, hip abductors, and knee extensors—to overall hip contact force were computed. Specific growth rate for the growth plate was computed, and the growth was simulated in the principal stress direction at each element in the growth front. The predicted growth indicated decreased NSA and FA (of about \(0.1 {^{\circ }}\) over a four-month period) for able-bodied children. Hip abductors contributed the most, and hip adductors, the least, to growth rate. All muscles groups contributed to a decrease in predicted NSA (\(\sim \)0.01\({^{\circ }}\)–0.04\({^{\circ }})\) and FA (\(\sim \)0.004\({^{\circ }}\)–\(0.2{^{\circ }}\)), except hip extensors and hip adductors, which showed a tendency to increase the FA (\(\sim \)0.004\({^{\circ }}\)–\(0.02{^{\circ }}\)). Understanding influences of different muscle groups on long bone growth tendency can help in treatment planning for growing children with affected gait.     

Stability of a Stochastic Model of an SIR Epidemic with Vaccination

We prove almost sure exponential stability for the disease-free equilibrium of a stochastic differential equations model of an SIR epidemic with vaccination. The model allows for vertical transmission. The stochastic perturbation is associated with the force of infection and is such that the total population size remains constant in time. We prove almost sure positivity of solutions. The main result concerns especially the smaller values of the diffusion parameter, and describes the stability in terms of an analogue \(\mathcal{R}_\sigma\) of the basic reproduction number \(\mathcal{R}_0\) of the underlying deterministic model, with \(\mathcal{R}_\sigma \le \mathcal{R}_0\). We prove that the disease-free equilibrium is almost sure exponentially stable if \(\mathcal{R}_\sigma <1\).     

A Mathematical Model of Anthrax Transmission in Animal Populations

A general mathematical model of anthrax (caused by Bacillus anthracis) transmission is formulated that includes live animals, infected carcasses and spores in the environment. The basic reproduction number \(\mathcal {R}_0\) is calculated, and existence of a unique endemic equilibrium is established for \(\mathcal {R}_0\) above the threshold value 1. Using data from the literature, elasticity indices for \(\mathcal {R}_0\) and type reproduction numbers are computed to quantify anthrax control measures. Including only herbivorous animals, anthrax is eradicated if \(\mathcal {R}_0 < 1\). For these animals, oscillatory solutions arising from Hopf bifurcations are numerically shown to exist for certain parameter values with \(\mathcal {R}_0>1\) and to have periodicity as observed from anthrax data. Including carnivores and assuming no disease-related death, anthrax again goes extinct below the threshold. Local stability of the endemic equilibrium is established above the threshold; thus, periodic solutions are not possible for these populations. It is shown numerically that oscillations in spore growth may drive oscillations in animal populations; however, the total number of infected animals remains about the same as with constant spore growth.