Modeling continuous glucose monitoring with fractional differential equations subject to shocks.

Continuous Glucose Monitoring (CGM) produces long time-series of noisy observations of a single variable (tissue glucose concentration), whose evolution may be explained by a dynamical model. In order to represent the unknown mixture of possible control mechanisms of different orders affecting the measured variable, a fractional differential approach seems justified. In any case, variations in food intake and/or physical activity ought to be taken into account if a plausible interpretation of the dynamics is to be obtained. In the present work, the mathematical construction and the numerical implementation of a Fractional Differential Equations (FDE) initial value problem are systematically reviewed, with the intent of offering the reader a concise and mathematically rigorous description of this approach. An FDE model for CGM is formulated: the model includes compartments for stomach and intestinal glucose contents and for blood and tissue (subcutaneous) glucose concentrations, as well as the shock effects of food ingestion and of increased glucose consumption due to physical activity. The model parameters, including the (non-integer) order of differentiation, are estimated from CGM observations on six Type 1 diabetic patients. The best-fit fractional orders for the six subjects range from 1.59 to 2.13. For comparison, best fits have also been computed for all subjects using an average fractional order of 1.9 and integer orders of 1 and 2.The results indicate that in the case of CGM the fractional differential model, which should be physiologically more appropriate, in fact fits the data much better than the first-order model and also better than the 2nd-order model.

Andronov-Hopf and Neimark-Sacker bifurcations in time-delay differential equations and difference equations with applications to models for diseases and animal populations.

In many areas, researchers might think that a differential equation model is required, but one might be forced to use an approximate difference equation model if data is only available at discrete points in time. In this paper, a detailed comparison is given of the behavior of continuous and discrete models for two representative time-delay models, namely a model for HIV and an extended logistic growth model. For each model, there are seven different time-delay versions because there are seven different positions to include time delays. For the seven different time-delay versions of each model, proofs are given of necessary and sufficient conditions for the existence and stability of equilibrium points and for the existence of Andronov-Hopf bifurcations in the differential equations and Neimark-Sacker bifurcations in the difference equations. We show that only five of the seven time-delay versions have bifurcations and that all bifurcation versions have supercritical limit cycles with one having a repelling cycle and four having attracting cycles. Numerical simulations are used to illustrate the analytical results and to show that critical times for Neimark-Sacker bifurcations are less than critical times for Andronov-Hopf bifurcations but converge to them as the time step of the discretization tends to zero.

Generalized reproduction numbers, sensitivity analysis and critical immunity levels of an SEQIJR disease model with immunization and varying total population size.

An SEQIJR model of epidemic disease transmission which includes immunization and a varying population size is studied. The model includes immunization of susceptible people (S), quarantine (Q) of exposed people (E), isolation (J) of infectious people (I), a recovered population (R), and variation in population size due to natural births and deaths and deaths of infected people. It is shown analytically that the model has a disease-free equilibrium state which always exists and an endemic equilibrium state which exists if and only if the disease-free state is unstable. A simple formula is obtained for a generalized reproduction number R g where, for any given initial population, R g < 1 means that the initial population is locally asymptotically stable and R g > 1 means that the initial population is unstable. As special cases, simple formulas are given for the basic reproduction number R 0 , a disease-free reproduction number R d f , and an endemic reproduction number R e n . Formulas are derived for the sensitivity indices for variations in model parameters of the disease-free reproduction number R d f and for the infected populations in the endemic equilibrium state. A simple formula in terms of the basic reproduction number R 0 is derived for the critical immunization level required to prevent the spread of disease in an initially disease-free population. Numerical simulations are carried out using the Matlab program for parameters corresponding to the outbreaks of severe acute respiratory syndrome (SARS) in Beijing, Hong Kong, Canada and Singapore in 2002 and 2003. From the sensitivity analyses for these four regions, the parameters are identified that are the most important for preventing the spread of disease in a disease-free population or for reducing infection in an infected population. The results support the importance of isolating infectious individuals in an epidemic and in maintaining a critical level of immunity in a population to prevent a disease from occurring.