A model of chemotaxis is analyzed that prevents blow-up of solutions. The model consists of a system of nonlinear partial differential equations for the spatial population density of a species and the spatial concentration of a chemoattractant in n-dimensional space. We prove the existence of solutions, which exist globally, and are L∞-bounded on finite time intervals. The hypotheses require nonlocal conditions on the species-induced production of the chemoattractant.

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

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

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

In this paper we explore a new model of field carcinogenesis, inspired by lung cancer precursor lesions, which includes dynamics of a spatially distributed population of pre-cancerous cells c(t, x), constantly supplied by an influx μ of mutated normal cells. Cell proliferation is controlled by growth factor molecules bound to cells, b(t, x). Free growth factor molecules g(t, x) are produced by precancerous cells and may diffuse before they become bound 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 cells or, mathematically, to stable spike solutions. These multiple lesions are consistent with the field theory of carcinogenesis. In a previous model published by these authors, the influx of mutated cells was equal to zero, μ = 0, which corresponded to a single pre-malignant colony of cells. In that model, stable patterns appeared only if some of the growth factor was 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 together allow understanding the dynamics of this model. The equilibrium solutions in the current model result from the balance between new premalignant colonies being initiated and the old ones dying out.

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

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

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

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

Mechanisms contributing to the spread of avian influenza seem to be well identified, but how their interplay led to the current worldwide spread pattern of H5N1 influenza is still unknown due to the lack of effective global surveillance and relevant data. Here we develop some deterministic models based on the transmission cycle and modes of H5N1 and focusing on the interaction among poultry, wild birds and environment. Some of the model parameters are obtained from existing literatures, and others are allowed to vary in order to assess the effectiveness of various control strategies involving bird migration, agro-ecological environments, live and dead poultry trading, smuggling of wild birds, mechanical movement of infected materials and specific farming practices. Our simulations are carried out for a set of parameters that leads to the basic reproduction number 3.3. We show that by reducing 95% of the initial susceptible poultry population or by killing all infected poultry birds within one day, one may control the disease outbreak in a local setting. Our simulation shows that cleaning the environment is also a feasible and useful control measure, but culling wild birds and destroying their habitat are ineffective control measures. We use a one dimensional PDE model to examine the contribution to the spatial spread rate by the size of the susceptible poultry birds, the diffusion rates of the wild birds and the virus. We notice the 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 the migration 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 often assumed to have exponentially distributed durations in epidemiological models. However, models that use the exponential distribution assumption (EDA) may generate biased and even misleading results in some cases. This discrepancy is particularly damaging if the models are employed to assist policy-makers in disease control and interventions. This paper studies a mathematical model that includes multiple infectious stages and general distributions for the stage durations (with the exponential distribution as a special case). Formulas for 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 are assumed. It is also shown that the disease dynamics are determined by the reproductive numbers.

Although age-related heterogeneity of infection has been addressed in various epidemic models assuming a demographically stationary population, only a few studies have explicitly 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 to account for the non-stationality of the host population. The basic reproduction number R0 is derived using the next generation operator, permitting discussions over the well-known invasion principles. The condition of endemic steady state is also characterized by using the effective next generation operator. Subsequently, estimators of R0 are offered which can explicitly account for non-zero population growth rate. Critical coverages of vaccination are also shown, highlighting the threshold condition for a population with varying size. When quantifying R0 using the force of infection estimated from serological data, it should be remembered that the estimate increases as the population growth rate decreases. On the contrary, given the same R0, critical coverage of vaccination in a growing population would be higher than that of decreasing population. Our exercise implies that high mass vaccination coverage at an early age would be needed to control childhood vaccine-preventable diseases in 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, we examine the impact that HIV-1 infection has on persons with latent tuberculosis. Based on previous work, we develop a mathematical model of an adaptive immune response in the lung which considers relevant immune effectors such as macrophages, various sub-populations of T-cells, and key cytokines to predict which mechanisms are important to HIV-1 infection induced reactivation of tuberculosis. Our results indicate that persons latently infected with TB who are subsequently co-infected with HIV-1 will suffer reactive TB. The mechanisms that contribute to this are essentially related to a completely different cytokine environment at the onset of HIV-1 infection due to the presence of Mycobacterium tuberculosis. Our analysis suggests that macrophages play an important role during co-infection and decreases in macrophage counts are coupled to a decline in CD4+ T-cells and increased viral loads. These mechanisms are also coupled to lower recruitment of T-cells and macrophages, compromising protective immunity in the lung and eventually leading to 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 treatment with 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 infection in the first place. The model considers susceptible, treated uninfected exposed, treated infected, (untreated) infectious, and recovered individuals. The overly optimistic assumptions are made that treated uninfected individuals are not susceptible and treated infected individuals are not infectious. Since treatment lengths are considered that have an arbitrary distribution, the model system consists of ordinary differential and integral equations. We study the impact of the treatment length distribution on the large-time behavior of the model solutions, namely whether the solutions converge to an equilibrium or whether they are driven into undamped oscillations.