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.

We consider two atomic transitions excited by two variable laser fields in a three-level system. We study the soliton-pair propagation out of resonance and under thermal bath effect. We present general analytical implicit expression of the soliton-pair shape. Furthermore, we show that when the coupling to the environment exceeds a critical value, the soliton-pair propagation through three-level atomic system will be prohibited.

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.

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.

In this review paper we consider physiologically structured population models that havebeen widely studied and employed in the literature to model the dynamics of a wide varietyof populations. However in a number of cases these have been found inadequate to describesome phenomena arising in certain real-world applications such as dispersion in thestructure variables due to growth uncertainty/variability. Prompted by this, we describedtwo recent approaches that have been investigated in the literature to describe thisgrowth uncertainty/variability in a physiologically structured population. One involvesformulating growth as a Markov diffusion process while the other entails imposing aprobabilistic structure on the set of possible growth rates across the entire population.Both approaches lead to physiologically structured population models with nontrivialdispersion. Even though these two approaches are conceptually quite different, they werefound in [17] to have a close relationship: in somecases with properly chosen parameters and coefficient functions, the resulting stochasticprocesses have the same probability density function at each time.

Biological rhythms occur at different levels in the organism. In single cells, the celldivision cycle shows rhythmicity in the way its molecular regulators, the cyclin dependantkinases (CDKs), modulate their activity periodically to ensure a healthy progression. Intissues, cell proliferation is driven by the circadian clock, which modulates theprogression through the cell cycle along the day. The circadian clock shows endogenousrhythmicity through a robust network of transcription-translation feedback loops thatcreate sustained oscillations. Rhythmicity is preserved in cell populations by thecoordination of the clocks among cells, through rhythmic synchronization signals. Here wediscuss mechanisms for generating rhythmic activities in cell populations by reviewingsome of the mathematical models that deal with them. We discuss the implication ofbiological rhythms for tissue growth and the possible application to chronomodulatedcancer treatments.

We introduce a model, similar to diffusion limited aggregation (DLA), which serves as adiscrete analog of the continuous dynamics of evaporation of thin liquid films. Withinmean field approximation the dynamics of this model, averaged over many realizations ofthe growing cluster, reduces to that of the idealized evaporation model in which surfacetension is neglected. However fluctuations beyond the mean field level play an importantrole, and we study their effect on the conserved quantities of the idealized evaporationmodel. Assuming the cluster to be a fractal, a heuristic approach is developed in order toexplain the distinctive increase of the fractal dimension with the cluster size.

A computational framework for testing the effects of cytotoxic molecules, specific to agiven phase of the cell cycle, and vascular disrupting agents (VDAs) is presented. Themodel is based on a cellular automaton to describe tumour cell states transitions fromproliferation to death. It is coupled with a model describing the tumour vasculature andits adaptation to the blood rheological constraints when alterations are induced by VDAstreatment. Several therapeutic protocols in two structurally different vascular networkswere tested by varying the duration of cytotoxic drug perfusion and the periodicity oftreatment cycles. The impact of VDAs were also tested both experimentally from intravitalmicroscopy through a dorsal skinfold chamber on a mouse and numerically. Simulationresults show that combining cytotoxic treatment with a post treatment of VDA through ajudicious timing could favour the rapid eradication of the tumour. The computationalframework thus gives some insights into the outcome of cytotoxic and VDAs treatments on aqualitative basis. Future validation from our experimental setup could open up newperspectives towards Computer-Assisted Therapeutic Strategies.

Cancer has recently overtaken heart disease as the world’s biggest killer. Cancer isinitiated by gene mutations that result in local proliferation of abnormal cells and theirmigration to other parts of the human body, a process called metastasis. The metastasizedcancer cells then interfere with the normal functions of the body, eventually leading todeath. There are two hundred types of cancer, classified by their point of origin. Most ofthem share some common features, but they also have their specific character. In thisarticle we review mathematical models of such common features and then proceed to describemodels of specific cancer diseases.

Influenza has been responsible for human suffering and economic burden worldwide. Isolation is one of the most effective means to control the disease spread. In this work, we incorporate isolation into a two-strain model of influenza. We find that whether strains of influenza die out or coexist, or only one of them persists, it depends on the basic reproductive number of each influenza strain, cross-immunity between strains, and isolation rate. We propose criteria that may be useful for controlling influenza. Furthermore, we investigate how effective isolation is by considering the host’s mean age at infection and the invasion rate of a novel strain. Our results suggest that isolation may help to extend the host’s mean age at infection and reduce the invasion rate of a new strain. When there is a delay in isolation, we show that it may lead to more serious outbreaks as compared to no delay.

In this paper we give a survey on modeling efforts concerning the CVRS. The material wediscuss is organized in accordance with modeling goals and stresses control and transportissues. We also address basic modeling approaches and discuss some of the challenges formathematical modeling methodologies in the context of parameter estimation and modelvalidation.

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.

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.

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 this paper we analyze the stochastic version of a minimalistic multi-strain model,which captures essential differences between primary and secondary infections in denguefever epidemiology, and investigate the interplay between stochasticity, seasonality andimport. The introduction of stochasticity is needed to explain the fluctuations observedin some of the available data sets, revealing a scenario where noise and complexdeterministic skeleton strongly interact. For large enough population size, the stochasticsystem can be well described by the deterministic skeleton gaining insight on the relevantparameter values purely on topological information of the dynamics, rather than classicalparameter estimation of which application is in general restricted to fairly simpledynamical scenarios.

We investigate optimal control of a cancer-immune cell interactive model with delay inthe interphase compartment. By applying the optimal control theory, we seek to minimizethe cost associated with the chemotherapy drug, minimize the accumulation of cancer cells,and increase the immune cell presence. Optimality conditions and characterization of thecontrol are provided. Numerical analyses are given to enhance the understanding of thedifficulties that occur in the control of cancer.

There is evidence that cancer develops when cells acquire a sequence of mutations thatalter normal cell characteristics. This sequence determines a hierarchy among the cells,based on how many more mutations they need to accumulate in order to become cancerous.When cells divide, they exhibit telomere loss and differentiate, which defines anothercell hierarchy, on top of which is the stem cell. We propose a mutation-generation model,which combines the mutation-accumulation hierarchy with the differentiation hierarchy ofthe cells, allowing us to take a step further in examining cancer development and growth.The results of the model support the hypothesis of the cancer stem cell’s role in cancerpathogenesis: a very small fraction of the cancer cell population is responsible for thecancer growth and development. Also, according to the model, the nature of mutationaccumulation is sufficient to explain the faster growth of the cancer cell population.However, numerical results show that in order for a cancer to develop within a reasonabletime frame, cancer cells need to exhibit a higher proliferation rate than normalcells.

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.