Connective stability of discontinuous large scale systems

AbstractThe stability of discontinuous large scale systems under structural perturbations are studied in this paper. It is assumed that the discontinuous equations possess solutions in the sense of Filippov. The results obtained yield sufficient conditions for connective stability. The interconnected systems are treated in terms of their subsystems

Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases

Hematopoiesis is a complex biological process that leads to the production and regulation of blood cells. It is based upon differentiation of stem cells under the action of growth factors. A mathematical approach of this process is proposed to carry out explanation on some blood diseases, characterized by oscillations in circulating blood cells. A system of three differential equations with delay, corresponding to the cell cycle duration, is analyzed. The existence of a Hopf bifurcation for a positive steady-state is obtained through the study of an exponential polynomial characteristic equation with delay-dependent coefficients. Numerical simulations show that long period oscillations can be obtained in this model, corresponding to a destabilization of the feedback regulation between blood cells and growth factors. This stresses the localization of periodic hematological diseases in the feedback loop

Projectors on the generalized eigenspaces for functional differential equations using integrated semigroups

AbstractThe aim of this article is to derive explicit formulas for the projectors on the generalized eigenspaces associated to some eigenvalues for linear functional differential equations (FDE) by using integrated semigroup theory. The idea is to formulate the FDE as a non-densely defined Cauchy problem and obtain an explicit formula for the integrated solutions of the non-densely defined Cauchy problem, from which we then derive explicit formulas for the projectors on the generalized eigenspaces associated to some eigenvalues. The results are useful in studying bifurcations in some semi-linear problems

Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics

We study a mathematical model describing the dynamics of a pluripotent stem cell population involved in the blood production process in the bone marrow. This model is a differential equation with a time delay. The delay describes the cell cycle duration and is uniformly distributed on an interval. We obtain stability conditions independent of the delay. We also show that the distributed delay can destabilize the entire system. In particularly, it is shown that Hopf bifurcations can occur

Modeling the Geographic Spread of Rabies in China

Abstract In order to investigate how the movement of dogs affects the geographically inter-provincial spread of rabies in Mainland China, we propose a multi-patch model to describe the transmission dynamics of rabies between dogs and humans, in which each province is regarded as a patch. In each patch the submodel consists of susceptible, exposed, infectious, and vaccinated subpopulations of both dogs and humans and describes the spread of rabies among dogs and from infectious dogs to humans. The existence of the disease-free equilibrium is discussed, the basic reproduction number is calculated, and the effect of moving rates of dogs between patches on the basic reproduction number is studied. To investigate the rabies virus clades lineages, the two-patch submodel is used to simulate the human rabies data from Guizhou and Guangxi, Hebei and Fujian, and Sichuan and Shaanxi, respectively. It is found that the basic reproduction number of the two-patch model could be larger than one even if the isolated basic reproduction number of each patch is less than one. This indicates that the immigration of dogs may make the disease endemic even if the disease dies out in each isolated patch when there is no immigration. In order to reduce and prevent geographical spread of rabies in China, our results suggest that the management of dog markets and trades needs to be regulated, and transportation of dogs has to be better monitored and under constant surveillance. Author Summary In 1999, human rabies cases were reported in about 120 counties in Mainland China, mainly in the southern provinces. Now outbreaks of human rabies have been reported in about 1000 counties and the disease has spread geographically from the south to the north. Phylogeographic analyses of rabies virus strains indicate that prevalent strains in northern provinces are indeed related to the remote southern provinces. It is believed that the geographical spread of rabies virus is caused by the transportation of dogs. In this paper, a multi-patch model is proposed to describe the spatial transmission dynamics of rabies in China and to investigate how the immigration of dogs affects the geographical spread of rabies. The expression and sensitivity analysis of the basic reproduction number indicates that the movement of dogs plays an essential role in the spatial transmission dynamics of rabies. Numerical simulations on the effect of the immigration rate in three pairs of provinces, Guizhou and Guangxi, Hebei and Fujian, Sichuan and Shaanxi, are also performed. It is shown that the immigration of dogs is the main factor for the long-distance inter-provincial spread of rabies and it is necessary to manage such inter-provincial transportation of dogs

Emergence and dynamics of influenza super-strains

Abstract Background Influenza super-strains can emerge through recombination of strains from birds, pigs, and humans. However, once a new recombinant strain emerges, it is not clear whether the strain is capable of sustaining an outbreak. In certain cases, such strains have caused major influenza pandemics. Methods Here we develop a multi-host (i.e., birds, pigs, and humans) and multi-strain model of influenza to analyze the outcome of emergent strains. In the model, pigs act as “mixing vessels” for avian and human strains and can produce super-strains from genetic recombination. Results We find that epidemiological outcomes are predicted by three factors: (i) contact between pigs and humans, (ii) transmissibility of the super-strain in humans, and (iii) transmissibility from pigs to humans. Specifically, outbreaks will reoccur when the super-strain intections are less frequent between humans (e.g., R0=1.4) but grequent from pigs to humans, and a large-scale outbreak followed by successively damping outbreaks will occur when human transmissibility is high (e.g., R0=2.3). The average time between the initial outbreak and the first resurgence varies from 41 to 82 years. We determine the largest outbreak will occur when 2.

Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay

AbstractThis paper is concerned with the existence, uniqueness and globally asymptotic stability of traveling wave fronts in the quasi-monotone reaction advection diffusion equations with nonlocal delay. Under bistable assumption, we construct various pairs of super- and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique nondecreasing traveling wave front (up to translation), which is monotonically increasing and globally asymptotically stable with phase shift. The influence of advection on the propagation speed is also considered. Comparing with the previous results, our results recovers and/or improves a number of existing ones. In particular, these results can be applied to a reaction advection diffusion equation with nonlocal delayed effect and a diffusion population model with distributed maturation delay, some new results are obtained