In this paper we present a theory describing the diffusion limited evaporation of sessilewater droplets in presence of contact angle hysteresis. Theory describes two stages ofevaporation process: (I) evaporation with a constant radius of the droplet base; and (II)evaporation with constant contact angle. During stage (I) the contact angle decreases fromstatic advancing contact angle to static receding contact angle, during stage (II) thecontact angle remains equal to the static receding contact angle. Universal dependencesare deduced for both evaporation stages. Obtained universal curves are validated againstavailable in the literature experimental data.

National policies regarding the BCG vaccine for tuberculosis vary greatly throughout the international community and several countries are currently considering discontinuing universal vaccination. Detractors of BCG point to its uncertain effectiveness and its interference with the detection and treatment of latent tuberculosis infection (LTBI).

In order to quantify the trade-off between vaccination and treatment of LTBI, a mathematical model was designed and calibrated to data from Brazil, Ghana, Germany, India, Mexico, Romania, the United Kingdom and the United States. Country-specific thresholds for when LTBI treatment outperforms mass vaccination were found and the consequences of policy changes were estimated.

Our results suggest that vaccination outperforms LTBI treatment in all settings but with greatly reduced efficiency in low incidence countries. While national policy statements emphasize BCG’s interference with LTBI detection, we find that reinfection should be more determinant of a country’s proper policy choice.

The influence of a global delayed feedback control which acts on a system governed by asubcritical complex Ginzburg-Landau equation is considered. The method based on avariational principle is applied for the derivation of low-dimensional evolution models.In the framework of those models, one-pulse and two-pulse solutions are found, and theirlinear stability analysis is carried out. The application of the finite-dimensional modelallows to reveal the existence of chaotic oscillatory regimes and regimes withdouble-period and quadruple-period oscillations. The diagram of regimes resembles thosefound in the damped-driven nonlinear Schrödinger equation. The obtained results arecompared with the results of direct numerical simulations of the original problem.

An eco-epidemiological model of susceptible Tilapia fish, infected Tilapia fish and Pelicans is investigated by several author based upon the work initiated by Chattopadhyay and Bairagi (Ecol. Model., 136, 103–112, 2001). In this paper, we investigate the dynamics of the same model by considering different parameters involved with the model as bifurcation parameters in details. Considering the intrinsic growth rate of susceptible Tilapia fish as bifurcation parameter, we demonstrate the period doubling route to chaos. Next we consider the force of infection as bifurcation parameter and demonstrate the occurrence of two successive Hopf-bifurcations. We identify the existence of backward Hopf-bifurcation when the death rate of predators is considered as bifurcation parameter. Finally we construct a stochastic differential equation model corresponding to the deterministic model to understand the role of demographic stochasticity. Exhaustive numerical simulation of the stochastic model reveals the large amplitude fluctuation in the population of fish and Pelicans for certain parameter values. Extinction scenario for Pelicans is also captured from the stochastic model.

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.

Consider the dynamics of a thin film flowing down an inclined plane under the action of gravity and in the presence of a first-order exothermic chemical reaction. The heat released by the reaction induces a thermocapillary Marangoni instability on the film surface while the film evolution affects the reaction by influencing heat/mass transport through convection. The main parameter characterizing the reaction-diffusion process is the Damköhler number. We investigate the complete range of Damköhler numbers. We analyze the steady state, its linear stability and nonlinear regime. In the latter case, long-wave models are compared with integral-boundary-layer ones and bifurcation diagrams for permanent solitary wave solutions of the different models are constructed. Time-dependent computations with the integral-boundary-layer models show that the system approaches a train of coherent structures that resemble the solitary pulses obtained in the bifurcation diagrams.

Mitochondria are one of the most important organelles determining Ca2+regulatory pathway in the cell. Recent experiments suggested the existence of cytosolicmicrodomains with locally elevated calcium concentration (CMDs) in the nearest vicinity ofthe outer mitochondrial membrane (OMM). These intermediate physical connections betweenendoplasmic reticulum (ER) and mitochodria are called MAM (mitochondria-associated ERmembrane) complexes.

The aim of this paper is to take into account the direct calcium flowfrom ER to mitochondria implied by the existence of MAMs and perform detailed numericalanalysis of the influence of this flow on the type and shape of calcium oscillations.Depending on the permeability of MAMs interface and ER channels, different patterns ofoscillations appear (simple, bursting and chaotic). For some parameters the oscillatorypattern disappear and the system tends to a steady state with extremely high calcium levelin mitochondria, which can be interpreted as a crucial point at the beginning of anapoptotic pathway.

RNA viruses replicate as complex and dynamic mutant distributions. They are termed viralquasispecies, in recognition of the fundamental contribution of quasispecies theory in ourunderstanding of error-prone replicative entities. Viral quasispecies have launched afertile field of transdiciplinary research, both experimental and theoretical. Here wereview the origin and some implications of the quasispecies concept, with emphasis oninternal interactions among components of the same mutant virus ensemble, a critical factto design new antiviral strategies. We make the distinction between “intrinsic” and“extrinsic” properties of mutant distributions, and emphasize that there are severallevels of complexity that can influence viral quasispecies behavior.

Tuberculosis (TB) remains a major global health problem. A possible risk factor for TB isdiabetes (DM), which is predicted to increase dramatically over the next two decades,particularly in low and middle income countries, where TB is widespread. This study aimedto assess the strength of the association between TB and DM. We present a deterministicmodel for TB in a community in order to determine the impact of DM in the spread of thedisease. The important mathematical features of the TB model are thoroughly investigated.The epidemic threshold known as the basic reproduction number and equilibria for the modelare determined and stabilities analyzed. The model is numerically analyzed to assess theimpact of DM on the transmission dynamics of TB. We perform sensitivity analysis on thekey parameters that drive the disease dynamics in order to determine their relativeimportance to disease transmission and prevalence. Numerical simulations suggest that DMenhances the TB transmission and progression to active TB in a community. The resultssuggest that there is a need for increased attention to intervention strategies such asthe chemoprophylaxis of TB latent individuals and treatment of active TB in people withDM, which may include testing for suspected diabetes, improved glucose control, andincreased clinical and therapeutic monitoring in order to reduce the burden of thedisease.

We review some recent results concerning Gibbs measures for nonlinear Schrödingerequations (NLS), with implications for the theory of the NLS, including stability andtypicality of solitary wave structures. In particular, we discuss the Gibbs measures ofthe discrete NLS in three dimensions, where there is a striking phase transition tosoliton-like behavior.

In order to better understand the dynamics of acute leukemia, and in particular to findtheoretical conditions for the efficient delivery of drugs in acute myeloblastic leukemia,we investigate stability of a system modeling its cell dynamics.

The overall system is a cascade connection of sub-systems consisting of distributeddelays and static nonlinear feedbacks. Earlier results on local asymptotic stability areimproved by the analysis of the linearized system around the positive equilibrium. For thenonlinear system, we derive stability conditions by using Popov, circle and nonlinearsmall gain criteria. The results are illustrated with numerical examples andsimulations.

Viruses are obligate intracellular parasites that rely on the host cell for expansion.With the development of global analyses techniques like transcriptomics, proteomics andsiRNA library screening of complete cellular gene sets, a large range of host cell factorshave been discovered that either support or restrict virus growth. Here we summarize someof the recent findings and focus our discussion on the hepatitis C virus and the humanimmunodeficiency virus, two major pathogens that threat global health. The identificationof cellular proteins affecting multiple viruses points to the existence of centralregulation nodes that might be exploited for both, a quantitative description ofhost-virus interactions within single infected cells and the development of novel,broad-spectrum antiviral drugs.

We present a unified mathematical approach to epidemiological models with parametricheterogeneity, i.e., to the models that describe individuals in the population as havingspecific parameter (trait) values that vary from one individuals to another. This is anatural framework to model, e.g., heterogeneity in susceptibility or infectivity ofindividuals. We review, along with the necessary theory, the results obtained using thediscussed approach. In particular, we formulate and analyze an SIR model with distributedsusceptibility and infectivity, showing that the epidemiological models for closedpopulations are well suited to the suggested framework. A number of known results from theliterature is derived, including the final epidemic size equation for an SIR model withdistributed susceptibility. It is proved that the bottom up approach of the theory ofheterogeneous populations with parametric heterogeneity allows to infer the populationlevel description, which was previously used without a firm mechanistic basis; inparticular, the power law transmission function is shown to be a consequence of theinitial gamma distributed susceptibility and infectivity. We discuss how the generaltheory can be applied to the modeling goals to include the heterogeneous contactpopulation structure and provide analysis of an SI model with heterogeneous contacts. Weconclude with a number of open questions and promising directions, where the theory ofheterogeneous populations can lead to important simplifications and generalizations.

A simple explicit numerical scheme is proposed for the solution of the Gardner–Ostrovskyequation (ut + cux + α uux + α1u2ux + βuxxx)x = γuwhich is also known as the extended rotation-modified Korteweg–de Vries(KdV) equation. This equation is used for the description of internal oceanic wavesaffected by Earth’ rotation. Particular versions of this equation with zero some ofcoefficients, α, α1, β, orγ are also known in numerous applications. The proposed numericalscheme is a further development of the well-known finite-difference scheme earlier usedfor the solution of the KdV equation. The scheme is of the second order accuracy both ontemporal and spatial variables. The stability analysis of the scheme is presented forinfinitesimal perturbations. The conditions for the calculations with the appropriateaccuracy have been found. Examples of calculations with the periodic boundary conditionsare presented to illustrate the robustness of the proposed scheme.

Existence and stability of periodic solutions are studied for a system of delaydifferential equations with two delays, with periodic coefficients. It models theevolution of hematopoietic stem cells and mature neutrophil cells in chronic myelogenousleukemia under a periodic treatment that acts only on mature cells. Existence of a guidingfunction leads to the proof of the existence of a strictly positive periodic solution by atheorem of Krasnoselskii. The stability of this solution is analysed.

The matrix KdV equation with a negative dispersion term is considered in the right upperquarter–plane. The evolution law is derived for the Weyl function of a correspondingauxiliary linear system. Using the low energy asymptotics of the Weyl functions, theunboundedness of solutions is obtained for some classes of the initial–boundaryconditions.

In this paper we are interested in a mathematical model of migration of grass eels in an estuary. We first revisit a previous model proposed by O. Arino and based on a degenerate convection-diffusion equation of parabolic-hyperbolic type with time-varying subdomains. Then, we propose an adapted mathematical framework for this model, we prove a result of existence of a weak solution and we propose some numerical simulations.

We consider the early carcinogenesis model originally proposed as a deterministicreaction-diffusion system. The model has been conceived to explore the spatial effectsstemming from growth regulation of pre-cancerous cells by diffusing growth factormolecules. The model exhibited Turing instability producing transient spatial spikes incell density, which might be considered a model counterpart of emerging foci of malignantcells. However, the process of diffusion of growth factor molecules is by its nature astochastic random walk. An interesting question emerges to what extent the dynamics of thedeterministic diffusion model approximates the stochastic process generated by the model.We address this question using simulations with a new software tool called sbioPN (spatialbiological Petri Nets). The conclusion is that whereas single-realization dynamics of thestochastic process is very different from the behavior of the reaction diffusion system,it is becoming more similar when averaged over a large number of realizations. The degreeof similarity depends on model parameters. Interestingly, despite the differences, typicalrealizations of the stochastic process include spikes of cell density, which however arespread more uniformly and are less dependent of initial conditions than those produced bythe reaction-diffusion system.

We obtain solvability conditions in H6(ℝ3) for asixth order partial differential equation which is the linearized Cahn-Hilliard problemusing the results derived for a Schrödinger type operator without Fredholm property in ourpreceding article [18].

The immune system is able to protect the host from tumor onset, and immune deficienciesare accompanied by an increased risk of cancer. Immunology is one of the fields in biologywhere the role of computational and mathematical modeling and analysis were recognized theearliest, beginning from 60s of the last century. We introduce the two most common methodsin simulating the competition among the immune system, cancers and tumor immunologystrategies: differential equations and rule-based models. Several specific implementationsare presented, describing in details how they work and how they advance or contribute thefield of tumor immunology.