Bifurcations and Stability in Models of Infectious Diseases

This work is concerned with bifurcation and stability in models related to various aspects of infections diseases. First, we study the dynamics of a mathematical model on primary and secondary cytotoxic T-lymphocyte responses to viral infections by Wodarz et al. This model has three equilibria and the stability criteria of them are discussed. We analytically show that periodic solutions may arise from the third equilibrium via Hopf bifurcation. Numerical simulations of the model agree with the theoretical results. These dynamical behaviours occur within biologically realistic parameter range. After studying the single-strain model, we analyze the bifurcation dynamics of an in vivo multi-strain model of Plasmodium falciparum. Main attention of this model is focused on the dynamics of cross-reactivity from antigenic variation. We apply the techniques of coupled cell systems to the model and find that synchrony-breaking Hopf bifurcation occurs from a nontrivial synchronous equilibrium. We prove the existence and calculate the condition for the bifurcation. Aside from studying the previous models, we also study the clustering behaviour found in numerous multi-strain transmission models. Numerical solutions of these models have shown that steady-state, periodic, or even chaotic motions can be self-organized into clusters. We show that the steady-state clusterings in existing models can be analytically predicted. The clusterings occur via semi-simple double zero bifurcations from the quotient network of the models. We also calculate the stability criteria for different clustering patterns. Finally, the biological implications of these results are discussed

The effects of symmetry on the dynamics of antigenic variation

In the studies of dynamics of pathogens and their interactions with a host immune system, an important role is played by the structure of antigenic variants associated with a pathogen. Using the example of a model of antigenic variation in malaria, we show how many of the observed dynamical regimes can be explained in terms of the symmetry of interactions between different antigenic variants. The results of this analysis are quite generic, and have wider implications for understanding the dynamics of immune escape of other parasites, as well as for the dynamics of multi-strain diseases.Comment: 21 pages, 4 figures; J. Math. Biol. (2012), Online Firs

Analysis of symmetries in models of multi-strain infections

In mathematical studies of the dynamics of multi-strain diseases caused by antigenically diverse pathogens, there is a substantial interest in analytical insights. Using the example of a generic model of multi-strain diseases with cross-immunity between strains, we show that a significant understanding of the stability of steady states and possible dynamical behaviours can be achieved when the symmetry of interactions between strains is taken into account. Techniques of equivariant bifurcation theory allow one to identify the type of possible symmetry-breaking Hopf bifurcation, as well as to classify different periodic solutions in terms of their spatial and temporal symmetries. The approach is also illustrated on other models of multi-strain diseases, where the same methodology provides a systematic understanding of bifurcation scenarios and periodic behaviours. The results of the analysis are quite generic, and have wider implications for understanding the dynamics of a large class of models of multi-strain diseases

On the stability and Hopf bifurcation of the non-zero uniform endemic equilibrium of a time-delayed malaria model

This article considers a time-delayed mathematical model of immune response to Plasmodium falciparum (Pf) malaria. Infected red blood cells display a wide variety of surface antigens to which the body in turn responds by mounting specific immune responses as well as cross-reactive immune responses. The model studied here tracks these infected red blood cells as well as the two types of immune responses. It is assumed that the immune responses are time-delayed, and hence a system of nonlinear delay differential equations is considered. The goal of the paper is to provide a vigorous analysis of the stability and Hopf bifurcation of the non-zero uniform endemic equilibrium of the mathematical model

Equivariant Hopf Bifurcation in a Time-Delayed Ring of Antigenic Variants

We consider an intrahost malaria model allowing for antigenic variation within a single species. The host’s immune response is compartmentalised into reactions to major and minor epitopes. We investigate the dynamics of the model, paying particular attention to bifurcation and stability of the uniform nonzero endemic equilibrium. We establish conditions for the existence of an equivariant Hopf bifurcation in a ring of antigenic variants, characterised by time delay

Clustering behaviour in networks with time delayed all-to-all coupling

Networks of coupled oscillators arise in a variety of areas. Clustering is a type of oscillatory network behavior where elements of a network segregate into groups. Elements within a group oscillate synchronously, while elements in different groups oscillate with a fixed phase difference. In this thesis, we study networks of N identical oscillators with time delayed, global circulant coupling with two approaches. We first use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to determine model independent existence and stability results for symmetric cluster solutions. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We then perform stability and bifurcation analysis to the original system of delay differential equations with symmetry. We first study the existence of Hopf bifurcations induced by coupling time delay, and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to two specfi c examples: a network of FitzHugh-Nagumo neurons with diffusive coupling and a network of Morris-Lecar neurons with synaptic coupling. In the case studies, we show how time delays in the coupling between neurons can give rise to switching between different stable cluster solutions, coexistence of multiple stable cluster solutions and solutions with multiple frequencies

Are biological systems poised at criticality?

Many of life's most fascinating phenomena emerge from interactions among many elements--many amino acids determine the structure of a single protein, many genes determine the fate of a cell, many neurons are involved in shaping our thoughts and memories. Physicists have long hoped that these collective behaviors could be described using the ideas and methods of statistical mechanics. In the past few years, new, larger scale experiments have made it possible to construct statistical mechanics models of biological systems directly from real data. We review the surprising successes of this "inverse" approach, using examples form families of proteins, networks of neurons, and flocks of birds. Remarkably, in all these cases the models that emerge from the data are poised at a very special point in their parameter space--a critical point. This suggests there may be some deeper theoretical principle behind the behavior of these diverse systems.Comment: 21 page

Phase Transitions Induced by Diversity and Examples in Biological Systems

Tesis leída en l'Universitat de les Illes Balears en diciembre de 2010The present thesis covers various topics that range over di erent aspects of scientific research. On one end there is the specific analysis of a precise form that models some experimental observations. A good theoretical understanding of the mathematics that describe the observations can be a guide to the experimentalist and help estimate the validity of the measurements. On the other end there are abstract models whose relation to physical systems seem far but they are prototypic for a broad range of di erent systems and the drawn conclusions tend to be quite general. Depending on the abstraction and on the simplifications in use the distinction between both ends might not be sharp. The ordering of the research results presented in part II of this thesis somehow reflects the seamless transition from one end to the other. To introduce the reader into the context of the genuine results we provide introductory material in the chapters of the present part I.Peer reviewe

Negative Feedback and Competition in the Yeast Polarity Establishment Circuit

<p>Many cells spontaneously establish a polarity axis even in the absence of directional cues, a process called symmetry breaking. A central question concerns how cells polarize towards one, and only one, randomly oriented "front". The conserved Rhotype GTPase Cdc42p is an essential factor for both directed and spontaneous polarization in various organisms, whose local activation is thought to define the cell's front. We previously proposed that in yeast cells, a small stochastic cluster of GTP-Cdc42p at a random site on the cortex can grow into a large, dominating cluster via a positive feedback loop involving the scaffold protein Bem1p. As stochastic Cdc42p clusters could presumably arise at many sites, why does only one site become the dominating "front"? We speculated that competition between growing clusters for limiting factors would lead to growth of a single winning "front" at the expense of the others. Utilizing time-lapse imaging with high spatiotemporal resolution, we now document initiation of multiple polarized clusters that competed rapidly to resolve a winning cluster. Such multicluster intermediates are observed in wild-type yeast cells with functional directional cues, but the locations where they are initiated are biased by the spatial cues. In addition, we detected an unexpected oscillatory polarization in a majority of the cells breaking symmetry, in which polarity factors initially concentrated very brightly and then dimmed in an oscillatory manner, dampening down to a final intermediate level after 2-3 peaks. Dampened oscillation suggests that the polarity circuit contains an in-built negative feedback loop. Mathematical modeling predicts that negative feedback would confer robustness to the polarity circuit and make the kinetics of competition between polarity factor clusters relatively insensitive to polarity factor concentration.</p><p>We are trying to understand how competition between clusters occurs. We find that the yeast guanine-nucleotide dissociation inhibitor (GDI), Rdi1p, is needed for rapid competition between clusters. In the absence of Rdi1p the initial clustering of polarity</p><p>factors is slowed, and competition is also much slower: in some cases cells still have two clusters at the time of bud emergence and they form two buds. We suggest that in the absence of Rdi1p, the clusters compete for a limiting pool of Cdc42p, and that slow</p><p>exchange of Cdc42p on and off the membrane in the absence of Rdi1p leads to slow competition.</p>Dissertatio