Contribution to the study of periodic chronic myelogenous leukemia

The period (in the order of 40 to 80 days) in periodic chronic myelogenous leukemia (PCML) oscillations is quite long compared with the duration of the cell cycle of the hematopoietic stem cells from which the oscillations are presumed to originate (in the order of one or two days). Our objective is to understand the origin of these long-period oscillations using a G0 model for stem cell dynamics. We determine the local stability conditions and show under what conditions the Hopf bifurcation may occur. We interpret the role of each parameter in the loss of stability, and then examine a simpler model to try to deduce possible changes at the stem-cell level that might be responsible for the characteristics PCML.     

HIV treatment models with time delay

The present paper shows possible effects of antiretroviral treatment on the dynamics of the spread of the disease of human immunodeficiency virus infection in a population of varying size. By introducing time delays, we model the latency period and the delayed onset of positive treatment effects in the patients. The Hopf bifurcation and stability behaviour of the delay differential-equation model are analysed and simulations for different scenarios depending on the size of the treatment-induced delay are presented, and the results are discussed in detail.     

Structured population on two patches: modeling dispersal and delay

We derive from the age-structured model a system of delay differential equations to describe the interaction of spatial dispersal (over two patches) and time delay (arising from the maturation period). Our model analysis shows that varying the immature death rate can alter the behavior of the homogeneous equilibria, leading to transient oscillations around an intermediate equilibrium and complicated dynamics (in the form of the coexistence of possibly stable synchronized periodic oscillations and unstable phase-locked oscillations) near the largest equilibrium.     

A simple SIS epidemic model with a backward bifurcation

It is shown that an SIS epidemic model with a non-constant contact rate may have multiple stable equilibria, a backward bifurcation and hysteresis. The consequences for disease control are discussed. The model is based on a Volterra integral equation and allows for a distributed infective period. The analysis includes both local and global stability of equilibria.     

Predicting adaptation of phenology in response to climate change, an insect herbivore example

Climate change has led to an advance in phenology in many species. Synchrony in phenology between different species within a food chain may be disrupted if an increase in temperature affects the phenology of the different species differently, as is the case in the winter moth egg hatch–oak bud burst system. Operophtera brumata (winter moth) egg hatch date has advanced more than Quercus robur (pedunculate oak) bud burst date over the past two decades. Disrupted synchrony will lead to selection, and a response in phenology to this selection may lead to species genetically adapting to their changing environment. However, a prerequisite for such genetic change is that there is sufficient genetic variation and severe enough fitness consequences. So far, examples of observed genetic change have been few. Using a half-sib design, we demonstrate here that O. brumata egg-hatching reaction norm is heritable, and that genetic variation exists. Fitness consequences of even a few days difference between egg hatch and tree bud opening are severe, as we experimentally determined. Estimates of genetic variation and of fitness were then combined with a climate scenario to predict the rate and the amount of change in the eggs' response to temperature. We predict a rapid response to selection, leading to a restoration of synchrony of egg hatch with Q. robur bud opening. This study shows that in this case there is a clear potential to adapt – rapidly – to environmental change. The current observed asynchrony is therefore not due to a lack of genetic variation and at present it is unclear what is constraining O. brumata to adapt. This kind of model may be particularly useful in gaining insight in the predicted amount and rate of change due to environmental changes, given a certain genetic variation and selection pressure.     

Population dynamics of a pathogen: the conundrum of vivax malaria

Building a mathematical model of population dynamics of pathogens within their host involves considerations of factors similar to those in ecology, as pathogens can prey on cells in the host. But within the multicellular host, attacked cell types are integrated with other cellular systems, which in turn intervene in the infection. For example, immune responses attempt to sense and then eliminate or contain pathogens, and homeostatic mechanisms try to compensate for cell loss. This review focuses on modeling applied to malarias, diseases caused by single-cell eukaryote parasites that infect red blood cells, with special concern given to vivax malaria, a disease often thought to be benign (if sometimes incapacitating) because the parasite only attacks a small proportion of red blood cells, the very youngest ones. However, I will use mathematical modeling to argue that depletion of this pool of red blood cells can be disastrous to the host if growth of the parasite is not vigorously check by host immune responses. Also, modeling can elucidate aspects of new field observations that indicate that vivax malaria is more dangerous than previously thought.     

Otolith formation,microstructure and daily increment validation in juvenile perch Perca fluviatilis

The otoliths of laboratory‐reared larval and juvenile perch Perca fluviatilis of known age were analysed to determine the age of otolith formation and validate the formation of daily increments. There was a linear relationship between number of increments and age in days post‐hatching, although by 82 days post‐hatching daily increment counts underestimated actual age by an average of 5 days. Otolith dimensions in relation to standard length indicated allometric growth of otoliths until completion of yolk absorption, and isometric growth thereafter, up to 82 days post‐hatching.     

Mean-square stability of a stochastic model for bacteriophage infection with time delays

We consider the stability properties of the positive equilibrium of a stochastic model for bacteriophage infection with discrete time delay. Conditions for mean-square stability of the trivial solution of the linearized system around the equilibrium are given by the construction of suitable Lyapunov functionals. The numerical simulations of the strong solutions of the arising stochastic delay differential system suggest that, even for the original non-linear model, the longer the incubation time the more the phage and bacteria populations can coexist on a stable equilibrium in a noisy environment for very long time.     

Dispersal delays, predator-prey stability, and the paradox of enrichment

It takes time for individuals to move from place to place. This travel time can be incorporated into metapopulation models via a delay in the interpatch migration term. Such a term has been shown to stabilize the positive equilibrium of the classical Lotka-Volterra predator-prey system with one species (either the predator or the prey) dispersing. We study a more realistic, Rosenzweig-MacArthur, model that includes a carrying capacity for the prey, and saturating functional response for the predator. We show that dispersal delays can stabilize the predator-prey equilibrium point despite the presence of a Type II functional response that is known to be destabilizing. We also show that dispersal delays reduce the amplitude of oscillations when the equilibrium is unstable, and therefore may help resolve the paradox of enrichment.