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.

In this survey, we briefly review some of our recent studies on predator-preymodels with discrete delay. We first study the distribution of zeros of a seconddegree transcendental polynomial. Then we apply the general results on thedistribution of zeros of the second degree transcendental polynomial to variouspredator-prey models with discrete delay, including Kolmogorov-typepredator-prey models, generalized Gause-type predator-prey models withharvesting, etc. Bogdanov-Takens bifurcations in delayed predator-prey modelswith nonmonotone functional response and in delayed predator-prey model withpredator harvesting are also introduced.

We study a reaction-diffusion equationwith an integral term describing nonlocal consumption of resourcesin population dynamics. We show that a homogeneous equilibrium canlose its stability resulting in appearance of stationary spatialstructures. They can be related to the emergence of biologicalspecies due to the intra-specific competition and randommutations. Various types of travelling waves are observed.

Hill functions follow from the equilibrium state of the reaction in which n ligands simultaneouslybind a single receptor. This result if often employed to interpret the Hill coefficient asthe number of ligand binding sites in all kinds of reaction schemes. Here, we study the equilibriumstates of the reactions in which n ligand bind a receptor sequentially, both non-cooperatively andin a cooperative fashion. The main outcomes of such analysis are that: n is not a good estimate,but only an upper bound, for the Hill coefficient; while the Hill coefficient depends quite stronglyon the cooperativity level among ligands. We finally use these results to discuss the feasibilityand constrains of using Hill functions to model the regulatory functions in mathematical models ofgene regulatory networks.

The newly developed unifying discontinuous formulation named the correction procedure viareconstruction (CPR) for conservation laws is extended to solve the Navier-Stokesequations for 3D mixed grids. In the current development, tetrahedrons and triangularprisms are considered. The CPR method can unify several popular high order methodsincluding the discontinuous Galerkin and the spectral volume methods into a more efficientdifferential form. By selecting the solution points to coincide with the flux points,solution reconstruction can be completely avoided. Accuracy studies confirmed that theoptimal order of accuracy can be achieved with the method. Several benchmark test casesare computed by solving the Euler and compressible Navier-Stokes equations to demonstrateits performance.

The inverse stable subordinator provides a probability model for time-fractional differential equations, and leads to explicit solution formulae. This paper reviews properties of the inverse stable subordinator, and applications to a variety of problems in mathematics and physics. Several different governing equations for the inverse stable subordinator have been proposed in the literature. This paper also shows how these equations can be reconciled.

We present a novel, cell-local shock detector for use with discontinuous Galerkin (DG)methods. The output of this detector is a reliably scaled, element-wise smoothnessestimate which is suited as a control input to a shock capture mechanism. Using anartificial viscosity in the latter role, we obtain a DG scheme for the numerical solutionof nonlinear systems of conservation laws. Building on work by Persson and Peraire, wethoroughly justify the detector’s design and analyze its performance on a number ofbenchmark problems. We further explain the scaling and smoothing steps necessary to turnthe output of the detector into a local, artificial viscosity. We close by providing anextensive array of numerical tests of the detector in use.

The cancer stem cell hypothesis has evolved to one of the most important paradigms inbiomedical research. During recent years evidence has been accumulating for the existenceof stem cell-like populations in different cancers, especially in leukemias. In thecurrent work we propose a mathematical model of cancer stem cell dynamics in leukemias. Weapply the model to compare cellular properties of leukemic stem cells to those of theirbenign counterparts. Using linear stability analysis we derive conditions necessary andsufficient for expansion of malignant cell clones, based on fundamental cellularproperties. This approach reveals different scenarios of cancer initiation and providesqualitative hints to possible treatment strategies.

We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus onphenomena related to unboundedness of the Laplacians. This includes (failure of) essentialselfadjointness, absence of essential spectrum and stochastic incompleteness.

One of the central results in Einstein’s theory of Brownian motion is that the mean square displacement of a randomly moving Brownian particle scales linearly with time. Over the past few decades sophisticated experiments and data collection in numerous biological, physical and financial systems have revealed anomalous sub-diffusion in which the mean square displacement grows slower than linearly with time. A major theoretical challenge has been to derive the appropriate evolution equation for the probability density function of sub-diffusion taking into account further complications from force fields and reactions. Here we present a derivation of the generalised master equation for an ensemble of particles undergoing reactions whilst being subject to an external force field. From this general equation we show reductions to a range of well known special cases, including the fractional reaction diffusion equation and the fractional Fokker-Planck equation.

We consider the spread of infectious disease through contact networks of ConfigurationModel type. We assume that the disease spreads through contacts and infected individualsrecover into an immune state. We discuss a number of existing mathematical models used toinvestigate this system, and show relations between the underlying assumptions of themodels. In the process we offer simplifications of some of the existing models. Thedistinctions between the underlying assumptions are subtle, and in many if not most casesthis subtlety is irrelevant. Indeed, under appropriate conditions the models areequivalent. We compare the benefits and disadvantages of the different models, and discusstheir application to other populations (e.g., clustered networks).Finally we discuss ongoing challenges for network-based epidemic modeling.

Basic models suitable to explain the epidemiology of dengue fever have previouslyshown the possibility of deterministically chaotic attractors, which might explain the observedfluctuations found in empiric outbreak data. However, the region of bifurcations and chaos requirestrong enhanced infectivity on secondary infection, motivated by experimental findings ofantibody-dependent-enhancement. Including temporary cross-immunity in such models, which iscommon knowledge among field researchers in dengue, we find bifurcations up to chaotic attractorsin much wider and also unexpected parameter regions of reduced infectivity on secondaryinfection, realistically describing more likely hospitalization on secondary infection when the viralload becomes high. The model shows Hopf bifurcations, symmetry breaking bifurcations of limitcycles, coexisting isolas, and two different possible routes to chaos, via the Feigenbaum perioddoubling and via torus bifurcations.

Cell-based, mathematical models helpmake sense of morphogenesis—i.e. cells organizing intoshape and pattern—by capturing cell behavior in simple, purelydescriptive models. Cell-based models then predict thetissue-level patterns the cells produce collectively. The firststep in a cell-based modeling approach is to isolatesub-processes, e.g. the patterning capabilities of one or afew cell types in cell cultures. Cell-based models can thenidentify the mechanisms responsible for patterning in vitro.This review discusses two cell culture models of morphogenesisthat have been studied using this combinedexperimental-mathematical approach: chondrogenesis (cartilagepatterning) and vasculogenesis (de novo blood vessel growth). Inboth these systems, radically different models can equallyplausibly explain the in vitro patterns. Quantitativedescriptions of cell behavior would help choose betweenalternative models. We will briefly review the experimentalmethodology (microfluidics technology and traction forcemicroscopy) used to measure responses of individual cells to theirmicro-environment, including chemical gradients, physical forcesand neighboring cells. We conclude by discussing how to includequantitative cell descriptions into a cell-based model: theCellular Potts model.

Theoretical studies and numerical experiments suggest that unstructured high-ordermethods can provide solutions to otherwise intractable fluid flow problems within complexgeometries. However, it remains the case that existing high-order schemes are generallyless robust and more complex to implement than their low-order counterparts. These issues,in conjunction with difficulties generating high-order meshes, have limited the adoptionof high-order techniques in both academia (where the use of low-order schemes remainswidespread) and industry (where the use of low-order schemes is ubiquitous). In this shortreview, issues that have hitherto prevented the use of high-order methods amongst anon-specialist community are identified, and current efforts to overcome these issues arediscussed. Attention is focused on four areas, namely the generation of unstructuredhigh-order meshes, the development of simple and efficient time integration schemes, th edevelopment of robust and accurate shock capturing algorithms, and finally the developmentof high-order methods that are intuitive and simple to implement. With regards to thisfinal area, particular attention is focused on the recently proposed flux reconstructionapproach, which allows various well known high-order schemes (such as nodal discontinuousGalerkin methods and spectral difference methods) to be cast within a single unifyingframework. It should be noted that due to the experience of the authors the review isdirected somewhat towards aerodynamic applications and compressible flow. However, many ofthe discussions have a wider applicability. Moreover, the tone of the review is intendedto be generally accessible, such that an extended scientific community can gain insightinto factors currently pacing the adoption of unstructured high-order methods.

Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusionshould be nonlinear if there exist non-diagonal terms. The vast variety of nonlinearmulticomponent diffusion equations should be ordered and special tools are needed toprovide the systematic construction of the nonlinear diffusion equations formulticomponent mixtures with significant interaction between components. We develop anapproach to nonlinear multicomponent diffusion based on the idea of the reaction mechanismborrowed from chemical kinetics.

Chemical kinetics gave rise to very seminal tools for the modeling of processes. This isthe stoichiometric algebra supplemented by the simple kinetic law. The results of thisinvention are now applied in many areas of science, from particle physics to sociology. Inour work we extend the area of applications onto nonlinear multicomponent diffusion.

We demonstrate, how the mechanism based approach to multicomponent diffusion can beincluded into the general thermodynamic framework, and prove the corresponding dissipationinequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementaryprocess cannot have an arbitrary form. For the general kinetic law (the generalized MassAction Law), additional conditions are proved. The cell–jump formalism gives anintuitively clear representation of the elementary transport processes and, at the sametime, produces kinetic finite elements, a tool for numerical simulation.

Tumor growth and progression is a complex phenomenon dependent on theinteraction of multiple intrinsic and extrinsic factors. Necessary for tumordevelopment is a small subpopulation of potent cells, so-called cancer stemcells, that can undergo an unlimited number of cell divisions and which areproposed to divide symmetrically with a small probability to produce more cancerstem cells. We show that the majority of cells in a tumor must indeed benon-stem cancer cells with limited life span and limited replicative potential. Tumor development is dependent as well on the proliferative potential and deathof these cells, and on the migratory ability of all cancer cells. Withincreasing number of cells in the tumor, competition for space limits tumorprogression, and in agreement with in vitro observation, the majority of cancercells become quiescent, with proliferation primarily occurring on the outer rimwhere space is available. We present an agent-based model of early tumordevelopment that captures the spatial heterogeneity of stemness andproliferation status. We apply the model to simulations of radiotherapy topredict treatment outcomes for tumors with different stem cell pool sizes anddifferent quiescence radiosensitivities. We show by first presuming homogeneousradiosensitivity throughout the tumor, and then considering the greaterresistance of quiescent cells, that stem cell pool size and stem cellrepopulation during treatment determine treatment success. The results fortumor cure probabilities comprise upper bounds, as there is evidence that cancerstem cells are also more radioresistant than other tumor cells. Beyond justdemonstrating the influence of mass effects of stem to non-stem cell ratios andproliferating to quiescent cell ratios, we show that the spatiotemporalevolution of the developing heterogeneous population plays a pivotal role indetermining radioresponse and treatment optimization.

While drawing a link between the papers contained in this issue and those present in a previous one (Vol. 2, Issue 3), this introductory article aims at putting in evidence some trends and challenges on cancer modelling, especially related to the development of multiphase and multiscale models.

The shearlet representation has gained increasing recognition in recent years as aframework for the efficient representation of multidimensional data. This representationconsists of a countable collection of functions defined at various locations, scales andorientations, where the orientations are obtained through the use of shear matrices. Whileshear matrices offer the advantage of preserving the integer lattice and being moreappropriate than rotations for digital implementations, the drawback is that the action ofthe shear matrices is restricted to cone-shaped regions in the frequency domain. Hence, inthe standard construction, a Parseval frame of shearlets is obtained by combiningdifferent systems of cone-based shearlets which are projected onto certain subspaces ofL2(ℝD) with the consequence thatthe elements of the shearlet system corresponding to the boundary of the cone regions losetheir good spatial localization property. In this paper, we present a new constructionyielding smooth Parseval frame of shearlets forL2(ℝD). Specifically, allelements of the shearlet systems obtained from this construction are compactly supportedand C∞ in the frequency domain, hence ensuring that the systemhas also excellent spatial localization.

We review some recent results for a class of fluid mechanics equations called activescalars, with fractional dissipation. Our main examples are the surface quasi-geostrophicequation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discussnonlocal maximum principle methods which allow to prove existence of global regularsolutions for the critical dissipation. We also recall what is known about the possibilityof finite time blow up in the supercritical regime.

In this work we study the inflammatory process resulting in the development of atherosclerosis. We develop a one- and two-dimensional models based on reaction-diffusion systems to describe the set up of a chronic inflammatory response in the intima of an artery vessel wall. The concentration of the oxidized low density lipoproteins (ox-LDL) in the intima is the critical parameter of the model. Low ox-LDL concentrations do not lead to a chronic inflammatory reaction. Intermediate ox-LDL concentrations correspond to a bistable system and can lead to a travelling wave propagation corresponding to a chronic inflammatory reaction. In this case the disease development depends on the initial condition. If the concentration of monocytes in the intima is sufficiently high, which can be caused by an inflammation related to other factors, then the development of atherosclerosis can start. Otherwise, the system returns to the stable disease free equilibrium. High ox-LDL concentrations correspond to a monostable system and even a small perturbation of the non inflammatory case leads to a travelling wave propagation which corresponds to a chronic inflammatory response.