In this work, we investigate a system of two nonlinear partial differential equations, arising from a model of cellular proliferation which describes the production of blood cells in the bone marrow. Due to cellular replication, the two partial differential equations exhibit a retardation of the maturation variable and a temporal delay depending on this maturity. Our aim is to prove that the behavior of primitive cells influences the global behavior of the population.
We consider a very general model of growth in the chemostat, where we suppose that the conversion between uptaken nutrient and cellular growth is variable and depends on the substrate concentration. We investigate some of the properties of this system when competition between several species is considered. In particular, we show numerically that competitor-mediated coexistence is possible, and present some of the very complex behaviors exhibited by this system.
A model for the dynamics of the production of white blood cells is derived and analysed. The model takes the form of a system of two delayed-differential equations with two discrete time lags. We identify the steady states and determine their stability. As the parameter values in the equations are allowed to vary, an equilibrium undergoes supercritical Hopf bifurcations, as well as saddle-node bifurcations of limit cycles. Care is taken to relate the bifurcations to the biological parameters inducing them. In particular, an increase in the apoptosis rate of either stem cells or white blood cell precursors is shown to be related to oscillations in the total number of circulating cells.
Joint work with Samuel Bernard and Michael Mackey.
Recurrent epidemics of infectious childhood diseases such as measles are a major health problem and have been subject to extensive theoretical research. Here we develop a theory for the dynamics of epidemic outbreaks and their synchronization in a network of coupled cities. Each city is described by a seasonally forced SEIR model. The model generates chaotic dynamics with annual and biennial dynamics in excellent agreement with long-term data sets. A new qualitative criterion based on the attractor topology is developed to distinguish between major outbreaks and epidemic fade-outs. This information is coded into a symbolic dynamics. We are able to deduce a one dimensional first return map of the chaotic SEIR equations, which upon iteration is able to generate the symbolic sequence of major outbreaks. The synchronization of epidemic outbreaks in a network of cities is defined as measure-based on the symbolic dynamics. This is applied to real data sets and numerical simulation results for different network topologies.
Disease transmission models with infectivity depending on the time since becoming infected were first formulated by W. O. Kermack and A. G. McKendrick [Proc. Roy. Soc. Ser. A 115(1927), 700-721]. However, variable infectivity was ignored until models for AIDS were developed by H. R. Thieme and C. Castillo-Chavez [SIAM J. Appl. Math. 53(1993), 1447 1479; Mathematical and Statistical Approaches to AIDS Epidemiology (C. Castillo-Chavez, ed.), Springer Verlag, 1989, 157-176]. We extend these models to models which include density-dependent demographics and possible recovery from infection. The central question is whether variable infectivity can cause instability of the endemic equilibrium.
Coupled loops with time delays are common in physiological systems such as neural networks. A Hopfield-type network is studied that consists of a pair of one-way loops each with three neurons with two-way coupling (of either excitatory or inhibitory type) between a single neuron of each loop. Time delays are introduced in the connections between the loops, and the effects of coupling strengths and delays on the network dynamics are investigated. It is shown that these effects depend strongly on whether the coupling is symmetric (of the same type in both directions) or asymmetric (inhibitory in one direction, excitatory in the other).
The objective of this research is to derive anatomically and physiologically accurate mathematical models of electrical activation in the human heart. These comprehensive simulation models will be used to
In this talk I will present some results exploring the effectiveness of quarantine strategies for newly emerging infectious diseases. Two issues will be addressed. In the first, I will use some simple probabilistic models to determine how quarantine durations should be set to minimize the risk that infected individuals are released back into the community. In the second I will ask whether quarantine is even a useful method for controlling emerging infectious diseases. This will be done through a comparison of the effectiveness of patient isolation in the presence and in the absence of quarantine, by using simple deterministic and stochastic models.
I briefly survey what is known about the pathogenesis of autoimmune (Type 1) diabetes, and how immune cells (CD8+ T cells) are triggered to proliferate and destroy the pancreatic cells (beta-cells) that produce insulin. I then summarize work done in my group on modeling a type of immunization (peptide therapy) procedure, and why caution has to be used in its application. This is joint work with A. F. M. Maree (Utrecht) and P. Santamaria (Calgary) and is funded by MITACS.
Industrial fishing has reduced the biomass of large predatory fish to about 10% of pre-fishing levels. But not all species exhibit a monotonic decline in abundance. For example, populations of Atlantic sailfish Istiophorus albicans often increase threefold before eventually being fished down to low levels. We construct nonlinear population models to understand these dynamics and to test various ecological hypotheses. We use state-space models in a Bayesian framework, which allows us to incorporate both observation error (the data are not precise) and process uncertainty (models are not exact representations of the real world). We utilise Markov Chain Monte Carlo (MCMC) methods, using the free software WinBUGS.
Human limbs deform because of many causes, including genetic predisposition, malnutrition, metabolic processes, diseases such as arthritis, and poor healing following fracture. Over the past seven years we have treated over 200 patients suffering from bone deformities, using custom software to plan and intraoperatively guide surgeons on complex reconstructive procedures.
This talk will present the principles of our work and clinical examples. It combines kinematics, dynamics, computer graphics, visualization, and 3D tracking to give surgeons unprecendented abilities to treat complex orthopedic conditions.
This talk will describe some recent work in modelling various forms of cancer treatment for different cancers. We think of cancer and normal cells as competing for bodily resources. Cancers at one site, several sites and throughout (such as leukemia) are considered. Chemotherapy, immunotherapy and radiation therapy models are described.
Models for assessing control strategies against the spread of HIV infection in a community as well as in vivo will be presented. The impact of the anti-HIV strategies in formulating an effective public health policy against HIV infection will be addressed. This is a collection of joint work with some members of the Mathematical Biology Team at the University of Manitoba.
A wide range of viral infections, such as HIV or influenza, can now be treated using antiviral drugs. Since viruses can evolve rapidly, the emergence and spread of drug resistant virus strains is a major concern. We shall describe within and between host models that can help indicate settings in which resistance is more or less likely to be problematic. In particular, we shall discuss the potential for the emergence of resistance in the context of human rhinovirus infection, an acute infection that is responsible for a large fraction of `common cold' cases.
We study a chemostat with two species feeding on two perfectly substitutable resources. The rate at which each species consumes each resource is assumed to be linear, and the growth yield ratios are assumed to be constant. Under certain conditions on the model parameters, Lyapunov functions can be used to demonstrate that there is a globally asymptotically stable equilibrium. Using the techniques of Li and Muldowney, the global behaviour can be determined for a larger subset of the parameter space. In particular, the global behaviour can be resolved for some cases for which the positive equilibrium is a saddle.
Joint work with Gail Wolkowicz and Mary Ballyk.
We consider the following single-species time delayed system in patchy
Our results show that at least one "food-rich" patch ensures permanence for the total system.
Distributions of dispersal times are incorporated into Lotka-Volterra models. These are formulated as integro-differential equations that describe predator-prey dynamics and dispersal between habitat patches. If one species disperses (predators are often more mobile than their prey), then dispersal almost always stabilizes the equilibrium. The exception occurs when every trip has exactly the same duration, thus the travel time distribution is a delta function. In this case of discrete delay, there is a set of parameter values for which the method used is inconclusive.
Joint work with Michael Neubert and Petra Klepac, Woods Hole Oceanographic Institute, USA.
Models for disease transmission in heterogeneous populations typically divide the population into several homogeneous compartments. Incidence of the disease is then due to contacts both within and between compartments. Nold (Math. Biosci. 52(1980), 227-240) proposed the following three models for mixing: proportional mixing, where contacts are made in proportion to the number of individuals in each compartment; restricted mixing, where contacts are strictly within compartments; and preferred mixing, a combination of the previous two. The models consist of a system of differential equations, with nonlinearities arising from bilinear (mass action) incidence terms and the coupling between compartments. We extend Nold's model to the case where contact are made in several settings. A quarantine/isolation model for the transmission of SARS-CoV is given as an example.
Some recent progress in the modeling and analysis of delayed spatial diffusion in structured populations will be presented. Model derivations will be discussed, and results on wave solutions, global attractors and synchronization will be reported.
During the transmission period of Severe Acute Respiratory Syndrome (SARS) we have predicted its future spread on the basis of epidemiological models and statistic data. The results were released in May 21, 2003, and the prediction matches the statistical date well. The simple SIR model is used for the prediction. The main attention is paid to the parameter estimation. An easy method is given to determine the transmission rate. The transmission rate is chosen as a time dependent parameter and has the shape of exponential curve to reflect the effect of various control measures. A prediction software is designed for the people who work in the public health departments. After the daily data of reported SARS case are input, the prediction curve can be given automatically. Few parameters are also introduced to show the influence of the start time and the stringency of the control measure on the transmission. Factors to be interweaved in epidemic modelling are mentioned.
By considering the mosquitos as with or without WNv and birds as infected and uninfected, I will introduce a set of differential equations to model the transmission of WNv among mosquitos and birds. Some analytical and numerical results will be presented.