A model of chemotaxis is analyzed that prevents blow-up of solutions. The modelconsists of a system of nonlinear partial differential equations for the spatial populationdensity of a species and the spatial concentration of a chemoattractant in n-dimensionalspace. We prove the existence of solutions, which exist globally, and are L∞-bounded onfinite time intervals. The hypotheses require nonlocal conditions on the species-inducedproduction of the chemoattractant.

We show how results by Diekmann et al. (2007) on the qualitative behaviour of solutionsof delay equations apply directly to a resource-consumer model with age-structured consumerpopulation.

Proteus mirabilis are bacteria that make strikingly regular spatial-temporal patterns on agar surfaces.In this paper we investigate a mathematical model that has beenshown to display these structures when solved numerically. The model consists of an ordinarydifferential equation coupled with a partial differential equation involving a first-orderhyperbolic aging term together with nonlinear degenerate diffusion. The system is shown toadmit global weak solutions.

Recently, adult stem cells have become a focus of intensive biomedical research, but thecomplex regulation that allows a small population of stem cells to replenish depleted tissues is stillunknown. It has been suggested that specific tissue structures delimit the spaces where stem cellsundergo unlimited proliferation (stem cell niche). In contrast, mathematical analysis suggests thata feedback control of stem cells on their own proliferation and differentiation (denoted QuorumSensing) suffices for preserving homeostasis in developing tissues. In the present work we examinethis discrepancy by analyzing a mathematical model of signalling pathways in a mammary stemcell. Our work shows that the model has a unique equilibrium depending on the environmentalconditions, which is stable for appropriate parameter values. We show how this dependence onthe micro-environment can control the cell decision to either proliferate or differentiate. Our workfurther suggests that this description is sufficient for the creation of niche conditions, with no needof additional structures. The analysis also pinpoints the pathways that may serve as the QuorumSensing mechanism.

In this paper we explore a new model of field carcinogenesis, inspired by lungcancer precursor lesions, which includes dynamics of a spatially distributed population ofpre-cancerous cells c(t, x), constantly supplied by an influx μ of mutated normal cells. Cellproliferation is controlled by growth factor molecules bound to cells, b(t, x). Free growthfactor molecules g(t, x) are produced by precancerous cells and may diffuse before they becomebound to other cells. The purpose of modelling is to investigate the existence of solutions,which correspond to formation of multiple spatially isolated lesions of pre-cancerous cellsor, mathematically, to stable spike solutions. These multiple lesions are consistent with thefield theory of carcinogenesis. In a previous model published by these authors, the influxof mutated cells was equal to zero, μ = 0, which corresponded to a single pre-malignantcolony of cells. In that model, stable patterns appeared only if some of the growth factorwas supplied from outside, arguably, a biologically tenuous hypothesis. In the present model,when μ > 0, that hypothesis is no more required, which makes this model more realistic.We present a range of results, both mathematical and computational, which taken togetherallow understanding the dynamics of this model. The equilibrium solutions in the currentmodel result from the balance between new premalignant colonies being initiated and theold ones dying out.

Recently the effect of a quiescent phase (or dormant/resting phase in applications) onthe dynamics of a system of differential equations has been investigated, in particular with respectto stability properties of stationary points. It has been shown that there is a general phenomenonof stabilization against oscillations which can be cast in rigorous form. Here we investigate, forhomogeneous systems, the effect of a quiescent phase, and more generally, a phase with slowerdynamics. We show that each exponential solution of the original system produces two exponentialsolutions of the extended system whereby the stability properties can be controlled.

In this paper several models in virus dynamics with and without immune response arediscussed concerning asymptotic behaviour. The case of immobile cells but diffusing viruses andT-cells is included. It is shown that, depending on the value of the basic reproductive number R0of the virus, the corresponding equilibrium is globally asymptotically stable. If R0 < 1 then thevirus-free equilibrium has this property, and in case R0 > 1 there is a unique disease equilibriumwhich takes over this property.

Let L be a nonsymmetric second order uniformly elliptic operator with generalWentzellboundary conditions. We show that a suitable version of L generates a quasicontractive semigroupon an Lp space that incorporates both the underlying domain and its boundary. This extends theearlier work of the authors on the symmetric case.

Hypercyclicity of C0-semigroups is a very unstable property: We give examples toshow that adding arbitrary small constants or a bounded rank one operator to the generator of ahypercyclic semigroup can destroy hypercyclicity. Also the limit of hypercyclic semigroups (evenin operator norm topology) need not be hypercyclic, and a hypercyclic semigroup can be the limitof nonhypercyclic ones. Hypercyclicity is not inherited by the Yosida approximations. Finally, therestriction of a hypercyclic nonnegative semigroup in a Banach lattice to the positive cone may befar from hypercyclic.

Mechanisms contributing to the spread of avian influenza seem to be well identified, buthow their interplay led to the current worldwide spread pattern of H5N1 influenza is still unknowndue to the lack of effective global surveillance and relevant data. Here we develop some deterministicmodels based on the transmission cycle and modes of H5N1 and focusing on the interactionamong poultry, wild birds and environment. Some of the model parameters are obtained fromexisting literatures, and others are allowed to vary in order to assess the effectiveness of variouscontrol strategies involving bird migration, agro-ecological environments, live and dead poultrytrading, smuggling of wild birds, mechanical movement of infected materials and specific farmingpractices. Our simulations are carried out for a set of parameters that leads to the basic reproductionnumber 3.3. We show that by reducing 95% of the initial susceptible poultry population or bykilling all infected poultry birds within one day, one may control the disease outbreak in a localsetting. Our simulation shows that cleaning the environment is also a feasible and useful controlmeasure, but culling wild birds and destroying their habitat are ineffective control measures. Weuse a one dimensional PDE model to examine the contribution to the spatial spread rate by thesize of the susceptible poultry birds, the diffusion rates of the wild birds and the virus. We noticethe diffusion rate of the wild birds with high mortality has very little impact on the spread speed.But for the wild birds who can survive the infection, depending on the direction of convection,their diffusion rate can substantially increase the spread rate, indicating a significant role of themigration of these type of wild birds in the spread of the disease.

Infectious diseases may have multiple infectious stages with very different epidemiological attributes,including infectivity and disease progression. These stages are oftenassumed to have exponentially distributed durations in epidemiological models. However,models that use the exponential distribution assumption (EDA) may generate biased andeven misleading results in some cases. This discrepancy is particularly damaging if the modelsare employed to assist policy-makers in disease control and interventions. This paperstudies a mathematical model that includes multiple infectious stages and general distributionsfor the stage durations (with the exponential distribution as a special case). Formulasfor the control reproductive number, Rc, and the basic reproductive number, R0, are derived,which can be conveniently applied to models in which specific stage distributions areassumed. It is also shown that the disease dynamics are determined by the reproductivenumbers.

Although age-related heterogeneity of infection has been addressed in variousepidemic models assuming a demographically stationary population, only a few studies haveexplicitly dealt with age-specific patterns of transmission in growing or decreasing population.To discuss the threshold principle realistically, the present study investigates an age-duration-structured SIR epidemic model assuming a stable host population, as the first scheme toaccount for the non-stationality of the host population. The basic reproduction number R0is derived using the next generation operator, permitting discussions over the well-knowninvasion principles. The condition of endemic steady state is also characterized by usingthe effective next generation operator. Subsequently, estimators of R0 are offered which canexplicitly account for non-zero population growth rate. Critical coverages of vaccination arealso shown, highlighting the threshold condition for a population with varying size. Whenquantifying R0 using the force of infection estimated from serological data, it should beremembered that the estimate increases as the population growth rate decreases. On thecontrary, given the same R0, critical coverage of vaccination in a growing population would behigher than that of decreasing population. Our exercise implies that high mass vaccinationcoverage at an early age would be needed to control childhood vaccine-preventable diseasesin developing countries.

Tuberculosis is the leading cause of death due to infectious diseases in the world today,and it is increasing due to co-infection with HIV-1, the causative agent of AIDS. Here, weexamine the impact that HIV-1 infection has on persons with latent tuberculosis. Based on previouswork, we develop a mathematical model of an adaptive immune response in the lung whichconsiders relevant immune effectors such as macrophages, various sub-populations of T-cells, andkey cytokines to predict which mechanisms are important to HIV-1 infection induced reactivationof tuberculosis. Our results indicate that persons latently infected with TB who are subsequentlyco-infected with HIV-1 will suffer reactive TB. The mechanisms that contribute to this are essentiallyrelated to a completely different cytokine environment at the onset of HIV-1 infection dueto the presence of Mycobacterium tuberculosis. Our analysis suggests that macrophages play animportant role during co-infection and decreases in macrophage counts are coupled to a decline inCD4+ T-cells and increased viral loads. These mechanisms are also coupled to lower recruitmentof T-cells and macrophages, compromising protective immunity in the lung and eventually leadingto TB reactivation. These results point to potential targets for drug and vaccine therapies.

A possible control strategy against the spread of an infectious disease is the treatmentwith antimicrobials that are given prophylactically to those that had contact with an infective person.The treatment continues until recovery or until it becomes obvious that there was no infectionin the first place. The model considers susceptible, treated uninfected exposed, treated infected,(untreated) infectious, and recovered individuals. The overly optimistic assumptions are made thattreated uninfected individuals are not susceptible and treated infected individuals are not infectious.Since treatment lengths are considered that have an arbitrary distribution, the model systemconsists of ordinary differential and integral equations. We study the impact of the treatment lengthdistribution on the large-time behavior of the model solutions, namely whether the solutions convergeto an equilibrium or whether they are driven into undamped oscillations.