The paper is concerned with an extension of the classical relation between the flame speed and the curvature-flow stretch, valid only for high Lewis numbers (diffusively stable flames). At low Lewis numbers the corresponding flame-flow system suffers short-wavelength instability, making the associated initial value problem ill-posed. In this study the difficulty is resolved by incorporation of higher-order effects. As a result one ends up with a reduced model based on a coupled system of second-order dynamic equations for the flame interface and its temperature. As an illustration the new model is applied for description of diffusively unstable stagnation-point flow flames.

Salmonella is one of the major sources of toxi-infection in humans in France and UnitedStates. The incidence of human salmonellosis has considerably increased over the past 20 years and this can be largely attributed to epidemics of S. enteritidis phage type 4 in poultry in numerous countries. In this article, we formulate and analyse a model in which the transmission of the disease is determined by contact between hens and Salmonella in the environment.

Since cancer is a complex phenomenon that incorporates events occurring on differentlength and time scales, therefore multiscale models are needed if we hope to adequately addresscancer specific questions. In this paper we present three different multiscale individual-cell-basedmodels, each motivated by cancer-related problems emerging from each of the spatial scales: extracellular,cellular or subcellular, but also incorporating relevant information from other levels.We apply these hybrid models to investigate the influence of the microenvironement on tumourinvasion, cell-cell collaboration and competition leading to the initiation and growth of epithelialtumours, and to evolution of cell phenotypes/genotypes arising in tumours growing in differentoxygen concentrations. We also discuss how these models relate to one another and can be used tobridge biological scales relevant to cancer.

The problem of finite-dimensional asymptotics of infinite-dimensional dynamicsystems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finite-dimensional asymptotics demonstrates effects of “natural” selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do nottend to fixed positions, and the path covered tends to infinity as t $\to$ ∞. The drift equationsfor peak motion are obtained. Various types of distribution stability are studied: internalstability (stability with respect to perturbations that do not extend the support), externalstability or uninvadability (stability with respect to strongly small perturbations that extendthe support), and stable realizability (stability with respect to small shifts and extensions ofthe density peaks). Models of self-synchronization of cell division are studied, as an exampleof selection in systems with additional symmetry. Appropriate construction of the notionof typicalness in infinite-dimensional space is discussed, and the notion of “completely thin”sets is introduced.

Mathematical models of tumour spheroids, proposed since the early seventies, have beengenerally formulated in terms of a single diffusive nutrient which is critical for cell replicationand cell viability. Only recently, attempts have been made to incorporate in the models the cellenergy metabolism, by considering the interplay between glucose, oxygen and lactate (or pH).By assuming glucose and lactate as the only fuel substrates, we propose a simple model for thecell ATP production which takes into account the main reactions that occur in the glycolytic andthe oxidative pathway. Under the assumption that cell death occurs when ATP production fallsbelow a critical level, we have studied the free boundary problem for the concentration of glucose,lactate and oxygen inside the spheroid viable rim. We show that the existence of a necrotic core isguaranteed for a sufficiently large size of the spheroid.

Whether the size of an animal population is environmentally limited or regulatedby density dependent negative feedback mechanisms is of ecological interest. Proponents oflimitation theory have issued a set of specific challenges which are addressed in this paperusing field data for the insect Tipula paludosa. This species is known to be subject to population crashes caused by adverse environmental conditions and assumed to be limited.We re-examine published data in support of this hypothesis and show that there is alsoevidence of negative feedback between generations. Meta-analyses of annual count data fromindividual fields, and regional surveys provide further evidence of negative feedback and itis concluded that population regulation occurs at both local and regional scales. Evidencefrom other studies is used to propose that cannibalism is the causative mechanism. Sincesimilar negative feedback responses were apparent under both normal and environmentallylimiting conditions when populations crash, a single population model that could simulatethe differing regional dynamics observed from the annual surveys was produced. We concludethat the range of annual fluctuations observed in T. paludosa populations is the outcome ofboth limiting and regulating processes, confirming from field observations that these are notexclusive mechanisms.

Simple epidemiological models with information dependent vaccination functions cangenerate sustained oscillations via Hopf bifurcation of the endemic state. The onset of these oscillations depend on the shape of the vaccination function. A “global” approach is used to characterize the instability condition and identify classes of functions that always lead to stability/instability. The analysis allows the identification of an analytically determined “threshold vaccination function” having a simple interpretation: coverage functions lying always above the threshold always lead to oscillations, whereas coverage functions always below never lead to instability.

The present work is a continuation of previous studies of premixed gas flames spreading through a space-periodic array of large-scale vorticities, and is motivated by the experimentally known phenomenon of flame extinction by turbulence. The prior work dealt with the strongly non-stoichiometric limit where the reaction rate is controlled by a single (deficient) reactant. In the present study the discussion is extended over a physically more realistic formulation based on a bimolecular reaction involving two reactants with different molecular diffusivities, and where the mixture equivalence ratio is utilized as a control param- eter. The flow-field is considered as prescribed and unaffected by combustion. The results obtained show that the dual influence of the large-scale turbulence on premixed combustion (flame speed enhancement followed by its reduction and extinction), and higher resilience of turbulent flames at lower Lewis numbers, are not actually related to the multiple-scale nature of the flow-field; the effects may well be captured within the framework of a one-scale flame-flow interaction scheme.


We study the influence of natural convection on stability of reaction fronts in porous media. The model consists of the heat equation, of the equation for the depth of conversion and of the equations of motion under the Darcy law. Linear stability analysis of the problem is fulfilled, the stability boundary is found. Direct numerical simulations are performed and compared with the linear stability analysis.

Seasonal forcing is identified as a key pattern generating mechanism in an epidemic model of rabies dispersal. We reduce an established individual-based high-detail modeldown to a deterministic conceptual model. The characteristic wave pattern characterized byhigh densities of infected individuals is maintained throughout the reduction process. In ourmodel it is evident that seasonal forcing is the dominant factor that drives pattern formation.In particular we show that seasonal forcing can cause splitting of epidemic wave fronts.

The aim of this paper is to study the steady states of the mathematical models with delaykernels which describe pathogen-immune dynamics of infectious diseases. In the study of mathematicalmodels of infectious diseases it is important to predict whether the infection disappearsor the pathogens persist. The delay kernel is described by the memory function that reflects theinfluence of the past density of pathogen in the blood and it is given by a nonnegative bounded andnormated function k defined on [ 0, ∞ ). By using the coefficient of the kernel k, as a bifurcationparameter, the models are found to undergo a sequence of Hopf bifurcation. The direction and the stability criteria of bifurcation periodic solutions are obtained by applying the normal form theory and the center manifold theorems. Some numerical simulation examples for justifying the theoretical results are also given.

Despite recent advances, treatment of patients with aggressive Non-Hodgkin'slymphoma (NHL2) has yet to be optimally designed. Notwithstanding the contribution ofmolecular treatments, intensification of chemotherapeutic regimens may still be beneficial.Hoping to aid in the design of intensified chemotherapy, we put forward a mathematicaland computational model that analyses the effect of Doxorubicin on NHL over a widerange of patho-physiological conditions. The model represents tumour growth both indiffusion-limited settings, that is, in small avascular tumours and tumour cords, and inperfusion-limited settings, e.g. in well-vascularized tumours. Model simulations indicatedthe presence of a critical regimen intensity below which treatment will fall short of tumourelimination. Taking this critical intensity into account, we compared two regimenintensification strategies: Dose escalation and regimen densification, i.e. reducing theinter-dosing interval. In the diffusion-limited setting, dose escalation was somewhat moreefficient than regimen densification. In the perfusion-limited setting, both intensificationstrategies yielded similar results. The present study coupled with a realistic myelotoxicitymodel may add insight on the optimisation of NHL intensified chemotherapy design.

We present a three species model describing the degradation of substrate bytwo competing populations of microorganisms in a marine sediment. Considering diffusionto be the main transport process, we obtain a reaction diffusion system (RDS) which westudy in terms of spontaneous pattern formation. We find that the conditions for patternsto evolve are likely to be fulfilled in the sediment. Additionally, we present simulationsthat are consistent with experimental data from the literature. We focus on the interactionbetween pattern formation and non-uniform forcing, which naturally occurs in our model,since layers closer to the sea water are richer in particular chemicals than deeper layers. We find that heterogeneous forcing has a strong impact on the structures that evolve, leading tothe formation of banded structures or even pulses instead of the well-known Turing patterns.

We give a survey of results on global stability for deterministic compartmental epidemiologicalmodels. Using Lyapunov techniques we revisit a classical result, and give a simple proof.By the same methods we also give a new result on differential susceptibility and infectivity modelswith mass action and an arbitrary number of compartments. These models encompass the so-calleddifferential infectivity and staged progression models. In the two cases we prove that if the basicreproduction ratio $\mathcal{R}_0$ ≤ 1, then the disease free equilibrium is globally asymptotically stable. If $\mathcal{R}_0$ > 1, there exists an unique endemic equilibrium which is asymptotically stable on the positive orthant.

A model of tumor growth in a spatial environment is analyzed. The model includesproliferating and quiescent compartments of tumor cells indexed by successively mutated cell phenotypesof increasingly proliferative aggressiveness. The model incorporates spatial dependencedue to both random motility and directed movement haptotaxis. The model structures tumor cellsby both cell age and cell size. The model consists of a system of nonlinear partial differentialequations for the compartments of tumor cells, extracellular matrix, matrix degradative enzyme,and oxygen. The existence, uniqueness, positivity, regularity, and growth characteristics of thesolutions are investigated.

A numerical study of a two-dimensional model for premixed gas combustion in a narrow, semi-infinite channel with no-slip boundary condition is performed. The work is motivated by recent theoretical advances revealing the major role of hydraulic resistance in deflagration-to-detonation transition, one of the central yet still inadequately understood phenomena of gaseous combustion. The work is a continuation and extension of recently reported results over non-isothermal boundary conditions, wider channels, and lower incipient flame velocities, closer to those of real life explosives.

In this paper we propose a mathematical model to describe the evolution of leukemiain the bone marrow. The model is based on a reaction-diffusion system of equations in a porousmedium. We show the existence of two stationary solutions, one of them corresponds to the normalcase and another one to the pathological case. The leukemic state appears as a result of a bifurcationwhen the normal state loses its stability. The critical conditions of leukemia developmentare determined by the proliferation rate of leukemic cells and by their capacity to diffuse. Theanalytical results are confirmed and illustrated by numerical simulations.

Gaseous detonation is a phenomenon with very complicated dynamics which has been studied extensively by physicists, mathematicians and engineers for many years. Despite many efforts the problem is far from a complete resolution. Recently the theory of subsonic detonation that occurs in highly resistant porous media was proposed in [4]. This theory provides a model which is realistic, rich and suitable for a mathematical study. In particular, the model is capable of describing the transition from a slowly propagating deflagration wave to the fast detonation wave. This phenomena is known as a deflagration to detonation transition and is one of the most challenging issues in combustion theory. In this paper we will present some recent mathematical results concerning initiation of reaction in porous media, existence and uniqueness of traveling fronts, quenching and propagation.

Previous work has shown that intracellular delay needs to be taken into account toaccurately determine the half-life of free virus from drug perturbation experiments [1]. The delayalso effects the estimated value for the infected T-cell loss rate when we assume that the drug isnot completely effective [19]. Models of virus infection that include intracellular delay are moreaccurate representations of the biological data.
We analyze a non-linear model of the human immunodeficiency virus (HIV) infection thatconsiders the interaction between a replicating virus, CD4+ T-cell and the cytotoxic-lymphocytes(CTL).We then investigate the intracellular delay effect on the stability of the endemically infectedsteady state. Criteria are given to ensure that the infected steady state is asymptotically stable forall delays. Model analysis also allows the prediction of a critical delay $\tau_c$ below which the effectorCTL can play a significant role in the immune control mechanism even when the basic reproductionnumber is high.

With a reaction-diffusion system, we consider the dispersing two-species Lotka-Volterra model with a temporally periodic interruption of the interspecific competitive relationship. We assume that the competition coefficient becomes a given positive constant and zero by turns periodically in time. We investigate the condition for the coexistence of two competing species in space, especially in the bistable case for the population dynamics without dispersion. We could find that the spatial coexistence, that is, the spatially mutual invasion of two competing species appears with two opposite-directed travelling waves if a condition for the temporal interruption of the interspecific relationship is satisfied. Further,we give a suggested mathematical expression of the velocity of travelling waves.