A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence

Explicit Lyapunov functions for SIR and SEIR compartmental epidemic models with nonlinear incidence of the form $\beta I_p S_q$ for the case p<1 are constructed. Global stability of the models is thereby established

Nonlinear incidence and stability of infectious disease models

In this paper we consider the impact of the form of the non-linearity of the infectious disease incidence rate on the dynamics of epidemiological models. We consider a very general form of the non-linear incidence rate (in fact, we assumed that the incidence rate is given by an arbitrary function f (S, I, N) constrained by a few biologically feasible conditions) and a variety of epidemiological models. We show that under the constant population size assumption, these models exhibit asymptotically stable steady states. Precisely, we demonstrate that the concavity of the incidence rate with respect to the number of infective individuals is a sufficient condition for stability. If the incidence rate is concave in the number of the infectives, the models we consider have either a unique and stable endemic equilibrium state or no endemic equilibrium state at all; in the latter case the infection-free equilibrium state is stable. For the incidence rate of the form g(I)h(S), we prove global stability, constructing a Lyapunov function and using the direct Lyapunov method. It is remarkable that the system dynamics is independent of how the incidence rate depends on the number of susceptible individuals. We demonstrate this result using a SIRS model and a SEIRS model as case studies. For other compartment epidemic models, the analysis is quite similar, and the same conclusion, namely stability of the equilibrium states, holds

Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains

Reaction-diffusion systems have been widely studied in developmental biology, chemistry and more recently in financial mathematics. Most of these systems comprise nonlinear reaction terms which makes it difficult to find closed form solutions. It therefore becomes convenient to look for numerical solutions: finite difference, finite element, finite volume and spectral methods are typical examples of the numerical methods used. Most of these methods are locally based schemes. We examine the implications of mesh structure on numerically computed solutions of a well-studied reaction-diffusion model system on two-dimensional fixed and growing domains. The incorporation of domain growth creates an additional parameter – the grid-point velocity – and this greatly influences the selection of certain symmetric solutions for the ADI finite difference scheme when a uniform square mesh structure is used. Domain growth coupled with grid-point velocity on a uniform square mesh stabilises certain patterns which are however very sensitive to any kind of perturbation in mesh structure. We compare our results to those obtained by use of finite elements on unstructured triangular elements

Modelling a new angle on understanding cancer

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Mathematical oncology: Cancer summed up [Concepts article]

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Cluster formation for multi-strain infections with cross-immunity

Many infectious diseases exist in several pathogenic variants, or strains, which interact via cross-immunity. It is observed that strains tend to self-organise into groups, or clusters. The aim of this paper is to investigate cluster formation. Computations demonstrate that clustering is independent of the model used, and is an intrinsic feature of the strain system itself. We observe that an ordered strain system, if it is sufficiently complex, admits several cluster structures of different types. Appearance of a particular cluster structure depends on levels of cross-immunity and, in some cases, on initial conditions. Clusters, once formed, are stable, and behave remarkably regularly (in contrast to the generally chaotic behaviour of the strains themselves). In general, clustering is a type of self-organisation having many features in common with pattern formation

Editorial

The last year has been a very busy one for the BMB. We have published 140 articles online, which is over twice the publication rate of the previous year. This reflects the increase in the submission rates and Springer's view that papers should be published online as quickly as possible. In addition, the entire archive of the Bulletin of Mathematical Biology is now available on the Springer website for the journal, digitized back to Vol. 1, No. 1, published in 1939

Phase differences in reaction-diffusion-advection systems and applications to morphogenesis

The authors study the effect of advection on reaction-diffusion patterns. It is shown that the addition of advection to a two-variable reaction–diffusion system with periodic boundary conditions results in the appearance of a phase difference between the patterns of the two variables which depends on the difference between the advection coefficients. The spatial patterns move like a travelling wave with a fixed velocity which depends on the sum of the advection coefficients. By a suitable choice of advection coefficients, the solution can be made stationary in time. In the presence of advection a continuous change in the diffusion coefficients can result in two Turing-type bifurcations as the diffusion ratio is varied, and such a bifurcation can occur even when the inhibitor species does not diffuse. It is also shown that the initial mode of bifurcation for a given domain size depends on both the advection and diffusion coefficients. These phenomena are demonstrated in the numerical solution of a particular reaction–diffusion system, and finally a possible application of the results to pattern formation in Drosophila larvae is discussed

Mathematical models for cell-matrix interactions during dermal wound healing

This paper contains a review of our recent work on the mathematical modeling of cell interaction with extracellular matrix components during the process of dermal wound healing. The models are of partial differential equation type and allow us to investigate in detail how various mechanochemical effects may be responsible for certain wound healing disorders such as fibrocontractive and fibroproliferative diseases. We also present a model for wound healing angiogenesis. The latter has several features in common with angiogenesis during cancer tumour growth and spread so a deeper understanding of the phenomenon in the context of wound healing may also help in the treatment of certain cancers

Episodic, transient systemic acidosis delays evolution of the malignant phenotype: Possible mechanism for cancer prevention by increased physical activity

Background\ud \ud The transition from premalignant to invasive tumour growth is a prolonged multistep process governed by phenotypic adaptation to changing microenvironmental selection pressures. Cancer prevention strategies are required to interrupt or delay somatic evolution of the malignant invasive phenotype. Empirical studies have consistently demonstrated that increased physical activity is highly effective in reducing the risk of breast cancer but the mechanism is unknown.\ud \ud Results\ud \ud Here we propose the hypothesis that exercise-induced transient systemic acidosis will alter the in situ tumour microenvironment and delay tumour adaptation to regional hypoxia and acidosis in the later stages of carcinogenesis. We test this hypothesis using a hybrid cellular automaton approach. This model has been previously applied to somatic evolution on epithelial surfaces and demonstrated three phases of somatic evolution, with cancer cells escaping in turn from the constraints of limited space, nutrient supply and waste removal. In this paper we extend the model to test our hypothesis that transient systemic acidosis is sufficient to arrest, or at least delay, transition from in situ to invasive cancer.\ud \ud Conclusions\ud \ud Model simulations demonstrate that repeated episodes of transient systemic acidosis will interrupt critical evolutionary steps in the later stages of carcinogenesis resulting in substantial delay in the evolution to the invasive phenotype. Our results suggest transient systemic acidosis may mediate the observed reduction in cancer risk associated with increased physical activity