Dynamically-consistent non-standard finite difference method for an epidemic model

This paper considers the problem of constructing finite-difference methods that are qualitatively consistent with the original continuous-time model they approximate. To achieve this goal, a deterministic continuous-time model for the transmission dynamics of two strains of an arbitrary disease, in the presence of an imperfect vaccine, is considered. The model is rigorously analysed, first of all, to gain insights into its dynamical features. The analysis reveal that it undergoes a vaccine-induced backward bifurcation when the associated reproduction threshold is less than unity. For the case where the vaccine is 100% effective, the disease-free equilibrium of the model is shown to be globally-asymptotically stable if the reproduction number is less than unity. The model also exhibits the phenomenon of competitive exclusion, where the strain with the higher reproduction number dominates (and drives out) the other. Two finite-difference methods are presented for numerically solving the model. The central objective is to determine which of the two methods gives solutions that are dynamically consistent with those of the continuous-time model. The first method, an implicitly-derived explicit finite-difference method, is considered for its computational simplicity, being a Gauss–Seidel-like algorithm. However, this method is shown to suffer numerous scheme-dependent numerical instabilities and spurious behaviour (such as convergence to the wrong steady-state solutions and failing to preserve many of the main essential dynamical features of the model), particularly when relatively large step-sizes are used in the simulations. On the other hand, the second numerical method, constructed based on Mickens’ non-standard finite-difference discretization framework, is shown to be free of any numerical instabilities and contrived behaviour regardless of the size of the step-size used in the numerical simulations. In other words, unlike the first method, the non-standard method is shown to be dynamically consistent with the original continuous-time model, and, therefore, it is more suited for use to study the asymptotic dynamics of the disease transmission model being considered.     

A second-order scheme for the “Brusselator” reaction–diffusion system

A second-order method is developed for the numerical solution of the initial-value problems , , and , , , in which the functions and , where A and B are positive real constants, are the reaction terms arising from the mathematical modelling of chemical systems such as in enzymatic reactions and plasma and laser physics in multiple coupling between modes. The method is based on three first-order methods for solving u and v, respectively. In addition to being second-order accurate in space and time, the method is seen to converge to the correct fixed point ( , V* = A/B) provided . The approach adopted is extended to solve a class of non-linear reaction–diffusion equations in two-space dimensions known as the “Brusselator” system. The algorithm is implemented in parallel using two processors, each solving a linear algebraic system as opposed to solving non-linear systems, which is often required when integrating non-linear partial differential equations (PDEs). This revised version was published online in July 2006 with corrections to the Cover Date.     

Dynamics analysis of a quarantine model in two patches

A new deterministic model for assessing the impact of quarantine on the transmission dynamics of a communicable disease in a two‐patch community is designed. Rigorous analysis of the model shows that the imperfect nature of quarantine (in the two patches) could induce the phenomenon of backward bifurcation when the associated reproduction number of the model is less than unity. For the case when quarantined susceptible individuals do not acquire infection during quarantine, the disease‐free equilibrium of the model is shown to be globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model has a unique Patch i‐only boundary equilibrium (i = 1,2) whenever the associated reproduction number for Patch i is greater than unity. The unique Patch i‐only boundary equilibrium is locally asymptotically stable whenever the invasion reproduction number of Patch 3 ? i is less than unity (and the associated reproduction number for Patch i exceeds unity). The model has at least one endemic equilibrium when its reproduction number exceeds unity (and the disease persists in both patches in this case). It is shown that adding multi‐patch dynamics to a single‐patch quarantine model (which allow the quarantine of susceptible individuals) in a single patch does not alter its quantitative dynamics (with respect to the existence and asymptotic stability of its associated equilibria as well as its backward bifurcation property). Copyright © 2014 John Wiley & Sons, Ltd.     

Lipase catalyzed ultrasonic synthesis of poly-4-hydroxybutyrate-co-6-hydroxyhexanoate

Four different lipases were compared for ultrasound-mediated synthesis of the biodegradable copolymer poly-4-hydroxybutyrate-co-6-hydroxyhexanoate. The copolymerization was carried out in chloroform. Of the enzymes tested, Novozym 435 exhibited the highest copolymerization rate, in fact the reaction rate was observed to increase with about 26-fold from 30 to 50 °C (7.9 × 10^?3 M s^?1), sonic power intensity of 2.6 × 10³ W m^?2 and dissipated energy of 130.4 J ml^?1. Copolymerization rates with the Candida antarctica lipase A, Candida rugosa lipase, and Lecitase Ultra? were lower at 2.4 × 10^?4, 1.3 × 10^?4 and 3.5 × 10^?4 M s^?1, respectively. The catalytic efficiency depended on the enzyme. The efficiency ranged from 4.15 × 10^?3 s^?1 M^?1 for Novozym 435–1.48 × 10^?3 s^?1 M^?1 for C. rugosa lipase. Depending on the enzyme and sonication intensity, the monomer conversion ranged from 8.2% to 48.5%. The sonication power, time and temperature were found to affect the rate of copolymerization. Increasing sonication power intensity from 1.9 × 10³ to 4.5 × 10³ W m^?2 resulted in an increased in acoustic pressure (P_a) from 3.7 × 10⁸ to 5.7 × 10⁸ N m^?2 almost 2.4–3.7 times greater than the acoustic pressure (1.5 × 10⁸ N m^?2) that is required to cause cavitation in water. A corresponding acoustic particle acceleration (a) of 9.6 × 10³–1.5 × 10⁴ m s^?2 was calculated i.e. approximately 984–1500 times greater than under the action of gravity.     

Radiological dose assessment of naturally occurring radioactive materials in concrete building materials

Previous studies have shown that the natural radioactivity contained in building materials have significantly influenced the dose rates in dwelling. Exposure to natural radiation in building has been of concerned since almost 80 % of our daily live are spend indoor. Thus, the aim of the study is to assess the radiological risk associated by natural radioactivity in soil based building materials to dwellers. A total of 13 Portland cement, 46 sand and 43 gravel samples obtained from manufacturers or bought directly from local hardware stores in Peninsular of Malaysia were analysed for their radioactivity concentrations. The activity concentrations of ²²⁶Ra, ²³²Th and ⁴⁰K in the studied building materials samples were found to be in the range of 3.7–359.3, 2.0–370.8 and 10.3–1,949.5 Bq kg^?1 respectively. The annual radiation dose rates (μSv year^?1) received by dwellers were evaluated for 1 to 50 years of exposure using Resrad-Build Computer Code based on the activity concentration of ²²⁶Ra, ²³²Th and ⁴⁰K found in the studied building material samples. The rooms modelling were based on the changing parameters of concrete wall thickness and the room dimensions. The annual radiation dose rates to dwellers were found to increase annually over a period of 50 years. The concrete thicknesses were found to have significantly influenced the dose rates in building. The self-absorption occurred when the concrete thickness was thicker than 0.4 m. Results of this study shows that the dose rates received by the dwellers of the building are proportional to the size of the room. In general the study concludes that concrete building materials; Portland cements, sands, and gravels in Peninsular of Malaysia does not pose radiological hazard to the building dwellers.     

Zeaxanthin Production by Novel Marine Isolates from Coastal sand of India and its Antioxidant Properties

Zeaxanthin carotenoids are class of commercially important natural products and diverse biomolecules produced by plants and many microorganisms. Bacteria often produce a cocktail of polar and nonpolar carotenoids limiting their industrial applications. Marine members of the family Flavobacteriaceae are known to produce potential carotenoids such as astaxanthin and zeaxanthin. A few bacterial species have been reported for the predominant production zeaxanthin. Here, we report the molecular identification of the zeaxanthin as a major carotenoid produced by two novel bacteria (YUAB-SO-11 and YUAB-SO-45) isolated from sandy beaches of South West Coast of India and the effect of carbon sources on the production of zeaxanthin. The strains were identified based on the 16S rRNA gene sequencing as a member of genus Muricauda. The closest relatives of YUAB-SO-11 and YUAB-SO-45 were Muricauda aquimarina (JCM 11811^T) (98.9 %) and Muricauda olearia (JCM 15563^T) (99.2 %), respectively, indicating that both of these strains might represent a novel species. The highest level of zeaxanthin production was achieved (YUAB-SO-11, 1.20?±?0.11 mg g^?1) and (YUAB-SO-45, 1.02?±?0.13 mg g^?1) when cultivated in marine broth supplemented with 2 % NaCl (pH 7) and incubated at 30 °C. Addition of 0.1 M glutamic acid, an intermediate of citric acid cycle, enhanced the zeaxanthin production as 18 and 14 % by the strains YUAB-SO-11 and YUAB-SO-45 respectively. The zeaxanthin showed in vitro nitric oxide scavenging, inhibition of lipid peroxidation, and 2,2-diphenyl-1-picryl hydrazyl scavenging activities higher than the commercial zeaxanthin. The results of this study suggest that two novel strains YUAB-SO-11 and YUAB-SO-45 belonging to genus Muricauda produce zeaxanthin as a predominant carotenoid, and higher production of zeaxanthin was achieved on glutamic acid supplementation. The pigment showed good in vitro antioxidant activity, which can be exploited further for commercial applications.     

Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis

A deterministic model for studying the transmission dynamics of bovine tuberculosis in a single cattle herd is presented and qualitatively analyzed. A notable feature of the model is that it allows for the importation of asymptomatically infected cattle (into the herd) because re‐stocking from outside sources. Rigorous analysis of the model shows that the model has a globally‐asymptotically stable disease‐free equilibrium whenever a certain epidemiological threshold, known as the reproduction number, is less than unity. In the absence of importation of asymptomatically infected cattle, the model has a unique endemic equilibrium whenever the reproduction number exceeds unity (this equilibrium is globally asymptotically stable for a special case). It is further shown that, for the case where asymptomatically infected cattle are imported into the herd, the model has a unique endemic equilibrium. This equilibrium is also shown to be globally asymptotically stable for a special case. Copyright © 2011 John Wiley & Sons, Ltd.     

Direct use of unassigned resonances in NMR structure calculations with proxy residues

We present a method that significantly enhances the robustness of (automated) NMR structure determination by allowing the NOE data corresponding to unassigned NMR resonances to be used directly in the calculations. The unassigned resonances are represented by additional atoms or groups of atoms that have no interaction with the regular protein atoms except through distance restraints. These so-called "proxy" residues can be used to generate NOE-based distance restraints in a similar fashion as for the assigned part of the protein. If sufficient NOE information is available, the restraints are expected to place the proxies at positions close to the correct atoms for the unassigned resonance, which can facilitate subsequent assignment. Convergence can be further improved by supplying additional information about the possible identities of the unassigned resonances. We have implemented this approach in the widely used automated assignment and structure calculation protocols ARIA and CANDID. We find that it significantly increases the robustness of structure calculations with regard to missing assignments and yields structures of higher quality. Our approach is still able to find correctly folded structures with up to 30% randomly missing resonance assignments, and even when only backbone and beta resonances are present! This should be of significant value to NMR-based structural proteomics initiatives.