A Fractional Order Recovery SIR Model from a Stochastic Process.

Over the past several decades, there has been a proliferation of epidemiological models with ordinary derivatives replaced by fractional derivatives in an ad hoc manner. These models may be mathematically interesting, but their relevance is uncertain. Here we develop an SIR model for an epidemic, including vital dynamics, from an underlying stochastic process. We show how fractional differential operators arise naturally in these models whenever the recovery time from the disease is power-law distributed. This can provide a model for a chronic disease process where individuals who are infected for a long time are unlikely to recover. The fractional order recovery model is shown to be consistent with the Kermack-McKendrick age-structured SIR model, and it reduces to the Hethcote-Tudor integral equation SIR model. The derivation from a stochastic process is extended to discrete time, providing a stable numerical method for solving the model equations. We have carried out simulations of the fractional order recovery model showing convergence to equilibrium states. The number of infecteds in the endemic equilibrium state increases as the fractional order of the derivative tends to zero.

Reaction-diffusion and reaction-subdiffusion equations on arbitrarily evolving domains.

Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a one-dimensional domain that is evolving. The model equations, which have been derived from generalized continuous time random walks, can incorporate complexities such as subdiffusive transport and inhomogeneous domain stretching and shrinking. Inhomogeneously growing domains are frequently encountered in biological phenomena involving stochastic transport, such as tumor growth and morphogen gradient formation. A method for constructing analytic expressions for short-time moments of the position of the particles is developed and moments calculated from this approach are shown to compare favorably with results from random walk simulations and numerical integration of the reaction transport equation. The results show the important role played by the initial condition. In particular, it strongly affects the time dependence of the moments in the short-time regime by introducing additional drift and diffusion terms. We also discuss how our reaction transport equation could be applied to study the spreading of a population on an evolving interface. From a more general perspective, our findings help to mitigate the scarcity of analytic results for reaction-diffusion problems in geometries displaying nonuniform growth. They are also expected to pave the way for further results, including the treatment of first-passage problems associated with encounter-controlled reactions in such domains.

Generalized fractional diffusion equations for subdiffusion in arbitrarily growing domains.

The ubiquity of subdiffusive transport in physical and biological systems has led to intensive efforts to provide robust theoretical models for this phenomena. These models often involve fractional derivatives. The important physical extension of this work to processes occurring in growing materials has proven highly nontrivial. Here we derive evolution equations for modeling subdiffusive transport in a growing medium. The derivation is based on a continuous-time random walk. The concise formulation of these evolution equations requires the introduction of a new, comoving, fractional derivative. The implementation of the evolution equation is illustrated with a simple model of subdiffusing proteins in a growing membrane.

Determination of the magnitude and enantioselectivity of ligand binding to rat and rabbit serum albumins using immobilized-protein high performance liquid chromatography stationary phases.

Rat, rabbit and human serum albumins were immobilized on an HPLC stationary phase, and the resulting phases were tested for their abilities to determine the extent and enantioselectivity of ligand binding to the respective albumins. A series of achiral and chiral compounds were chromatographed on the phases including benzodiazepinones, non-steroidal anti-inflammatory drugs, amino acids, warfarin and leucovorin. The chromatographic retentions of the benzodiazepinones and one series of nonsteroidal anti-inflammatory agents were compared with protein binding data from ultrafiltration studies. The observed correlation factors (r) were consistently 0.999, indicating that the albumin phases can be used to determine the magnitude of binding to the respective proteins. The enantioselectivity was also investigated, and the results indicate that the stationary phases can be used to determine relative enantioselectivities and intraspecies differences in this stereoselectivity. For example, when R- and S-warfarin were studied, R-warfarin was retained to a greater extent than S-warfarin by the rabbit serum albumin-stationary phase, whereas the opposite enantioselectivity was found for the rat and human albumins. Binding interaction studies were also conducted on the rabbit and rat albumin stationary phases by sequentially adding increasing concentrations of octanoic acid to the chromatographic mobile phase. The octanoic acid reduced the retention of a series of non-steroidal anti-inflammatory agents, and the results of the experiments suggest that the interaction takes place at two or more sites on the albumin molecule and by anti-cooperative allosteric interactions and competitive displacement. The results of this study demonstrate that the immobilized serum albumin columns can be used to quantitate and probe ligand binding interactions.

Abnormalities of carbohydrate metabolism and of OCT gene function in the Rett syndrome.

The pathogenetic basis of the Rett syndrome (RS) is unknown: an X-linked dominant, male-lethal gene defect is thought likely. We present a girl with RS who has defects both of the urea cycle and of carbohydrate metabolism resulting in fasting hypoglycaemia, post-prandial hyperlactataemia and excess urinary orotic acid excretion after alanine load. Her sister has a similar clinical picture, but less marked metabolic anomalies. The mother of these sisters has abnormal urinary orotic acid excretion; she transmitted opposite ornithine carbomoyltransferase (OCT) alleles to the two girls. Another girl with RS has similar metabolic responses to fasting and to carbohydrate load. We conclude that RS may be an aetiologically homogeneous condition, but that it includes a variable pattern of metabolic anomalies, and that the gene defect is distinct from the OCT locus.

Comparison of drug binding interactions on human, rat and rabbit serum albumin using high-performance displacement chromatography.

The binding of warfarin, a series of non-steroidal anti-inflammatory drugs and a series of benzodiazepines to rat serum albumin (RatSA) and rabbit serum albumin (RabSA) was compared with their binding to human serum albumin (HSA) using high-performance liquid chromatography on stationary phases based on immobilized albumins. The effect of the addition to the mobile phase of compounds known to bind to HSA at site I (phenylbutazone) or at site II (R- and S-ibuprofen) or at both sites (2,3,5-triiodobenzoic acid) was investigated on all three proteins. The results indicated that for the chiral compounds studied, the stereoselectivity of drug binding was much lower on RatSA than on HSA. On RatSA and RabSA, the benzodiazepine site was not a major binding site for R- and S-ibuprofen. The results indicated the existence of two binding sites for R and S warfarin on RatSA and probably on RabSA. On RatSA, one site is the major stereoselective site and is the major binding site of phenylbutazone and piroxicam. The other one is a major binding site for R- and S-ibuprofen and R- and S-ketoprofen.

Evaluation of beta-lactoglobulin as a stationary phase in high-performance liquid chromatography and as a buffer additive in capillary electrophoresis: observation of a surprising lack of stereoselectivity.

Previous studies have reported that alpha1-acid glycoprotein is quite similar in amino acid sequence and disulfide bond arrangements to members of a group of proteins which include beta-lactoglobulin (BLG). Since generally homologous proteins retain some similarity in function at the molecular level, we decided to evaluate the enantioselective properties of BLG as an high-performance liquid chromatographic chiral stationary phase (HPLC-CSP), and as an additive in capillary electrophoresis (CE). Two columns with differences in internal diameter and method of immobilisation on epoxide silica were prepared. Chiral acidic, basic and uncharged drugs were chromatographed and mobile phase parameters, namely pH and type of organic modifier, were varied in order to test the column performance. The CE approach has some advantages in that there is no need for immobilisation and only a small amount of protein is required. BLG was therefore tested as a CE buffer additive, using the same analytes as in the HPLC study. Although one would expect that a protein would display some enantioselectivity, BLG did not show any enantioselectivity whatsoever in either system; the protein has fairly weak interaction with the majority of the test solutes, as indicated by both techniques.

A practical method of serial bedside measurement of cerebral blood flow and metabolism during neurointensive care.

Acute encephalopathy is a major cause of death and neurological handicap in children. The principle aims of treatment are to provide adequate cerebral blood flow for the brain's metabolic needs and to prevent intracranial pressure rising above the level at which brain herniation occurs. Rational management requires an understanding of the pathophysiological changes in cerebral blood flow and metabolism which occur. The paucity of data on this subject reflects the perceived difficulty of measuring cerebral blood flow and cerebral metabolism in children. A modification of the Kety Schmidt technique of measuring cerebral blood flow and cerebral metabolism is described. This modification makes it possible to perform serial bedside measurements in children receiving intensive care. This method was used to perform 348 measurements in 58 children. The method was reproducible and no significant complications were encountered. The results indicated that appreciable changes in cerebral blood flow and metabolism could occur in individual patients over time, emphasising the importance of serial measurements. This technique may provide a practical means of monitoring cerebral blood flow and metabolism in very sick children receiving neurointensive care and evaluating the efficacy of treatment.