Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model

In this paper we consider the phenomenon of backward bifurcation in epidemic modelling illustrated by an extended model for Bovine Respiratory Syncytial Virus (BRSV) amongst cattle. In its simplest form, backward bifurcation in epidemic models usually implies the existence of two subcritical endemic equilibria for R 0 < 1, where R 0 is the basic reproductive number, and a unique supercritical endemic equilibrium for R 0 > 1. In our three-stage extended model we find that more complex bifurcation diagrams are possible. The paper starts with a review of some of the previous work on backward bifurcation then describes our three-stage model. We give equilibrium and stability results, and also provide some biological motivation for the model being studied. It is shown that backward bifurcation can occur in the three-stage model for small b, where b is the common per capita birth and death rate. We are able to classify the possible bifurcation diagrams. Some realistic numerical examples are discussed at the end of the paper, both for b small and for larger values of b

Gradients versus Cycling in Genetic Selection Models

We review the hierarchy of (continuous time) selection models starting with the classical Fisher's viability selection model, and its generalizations when allowing mutations, recombination, sex-dependent viabilities, fertility selection and different mortality rates. We analyse the question in which way Fisher's "Fundamental Theorem of Natural Selection" and Kimura's Maximum Principle can be extended to these more general situations. It turns out that in many cases this is principally impossible since the dynamics becomes cycling or even chaotic

Oscillations in low-dimensional cyclic differential delay systems

Nonlinear autonomous N-dimensional systems of cyclic differential equations with delays and overall negative feedback are considered. Such systems serve as mathematical models of numerous real world phenomena in physics and laser optics, physiology and mathematical biology, economics and life sciences among others. In the case of lower dimensions and sufficient conditions are derived for the oscillation of all solutions about the unique equilibrium. Open problems and conjectures are discussed for the higher dimensional case and for more convoluted sign feedbacks. © 2018, Springer Nature Switzerland AG

Winner-take-all selection in a neural system with delayed feedback

We consider the effects of temporal delay in a neural feedback system with excitation and inhibition. The topology of our model system reflects the anatomy of the avian isthmic circuitry, a feedback structure found in all classes of vertebrates. We show that the system is capable of performing a `winner-take-all' selection rule for certain combinations of excitatory and inhibitory feedback. In particular, we show that when the time delays are sufficiently large a system with local inhibition and global excitation can function as a `winner-take-all' network and exhibit oscillatory dynamics. We demonstrate how the origin of the oscillations can be attributed to the finite delays through a linear stability analysis.Comment: 8 pages, 6 figure

Pushed and pulled fronts in a discrete reaction-diffusion equation

We consider the propagation of wave fronts connecting unstable and stable uniform solutions to a discrete reaction-diffusion equation on a one-dimensional integer lattice. The dependence of the wavespeed on the coupling strength µ between lattice points and on a detuning parameter (α) appearing in a nonlinear forcing is investigated thoroughly. Via asymptotic and numerical studies, the speed both of 'pulled' fronts (whereby the wavespeed can be characterised by the linear behaviour at the leading edge of the wave) and of 'pushed' fronts (for which the nonlinear dynamics of the entire front determine the wavespeed) is investigated in detail. The asymptotic and numerical techniques employed complement each other in highlighting the transition between pushed and pulled fronts under variations of µ and α

Age groups and spread of influenza: implications for vaccination strategy

<p>Abstract</p> <p>Background</p> <p>The unpredictable nature of the potentially devastating impact of 2009 pH1N1 influenza pandemic highlights the need for pandemic preparedness planning, where modeling studies could be most useful for simulations of possible future scenarios.</p> <p>Methods</p> <p>A compartmental model with pre-symptomatic and asymptomatic influenza infections is proposed which incorporates age groups as well as intervention measures such as age-specific vaccination, in order to study spread of influenza in a community.</p> <p>Results</p> <p>We derive the basic reproduction number and other effective reproduction numbers under various intervention measures. For illustration, we make use of the Pneumonia and Influenza (P&I) mortality data and vaccination data of the very young (age 0-2) and the very old (age >64) during 2004-2005 Taiwan winter influenza season to fit our model and to compute the relevant reproduction numbers. The reproduction number for this winter flu season is estimated to be slightly above one (~1.0001).</p> <p>Conclusions</p> <p>Comparatively large errors in fitting the P&I mortality data of the elderly (>64) were observed shortly after winter school closings in January, which may indicate the impact of younger, more active age groups transmitting influenza to other age groups outside of the school settings; in particular, to the elderly in the households. Pre-symptomatic infections seemed to have little effect on the model fit, while asymptomatic infection by asymptomatic infectives has a more pronounced impact on the model fit for the elderly mortality, perhaps indicating a larger role in disease transmission by asymptomatic infection. Simulations indicate that the impact of vaccination on the disease incidence might not be fully revealed in the change (or the lack thereof) in the effective reproduction number with interventions, but could still be substantial. The estimated per contact transmission probability for susceptible elderly is significantly higher than that of any other age group, perhaps highlighting the vulnerability of the elderly due to close contacts with their caretakers from other age groups. The relative impact of targeting the very young and the very old for vaccination was weakened by their relative inactivity, thus giving evidence of the lack of impact of vaccinating these two groups on the overall transmissibility of the disease in the community. This further underscores the need for morbidity-based strategy to prevent elderly mortality.</p

Effect of Heterogeneous Mixing and Vaccination on the Dynamics of Anthelmintic Resistance: A Nested Model

Anthelmintic resistance is a major threat to current measures for helminth control in humans and animals. The introduction of anthelmintic vaccines, as a complement to or replacement for drug treatments, has been advocated as a preventive measure. Here, a computer-based simulation, tracking the dynamics of hosts, parasites and parasite-genes, shows that, depending on the degree of host-population mixing, the frequency of totally recessive autosomes associated with anthelmintic resistance can follow either a fast dynamical regime with a low equilibrium point or a slow dynamical regime with a high equilibrium point. For fully dominant autosomes, only one regime is predicted. The effectiveness of anthelminthic vaccines against resistance is shown to be strongly influenced by the underlying dynamics of resistant autosomes. Vaccines targeting adult parasites, by decreasing helminth fecundity or lifespan, are predicted to be more effective than vaccines targeting parasite larvae, by decreasing host susceptibility to infection, in reducing the spread of resistance. These results may inform new strategies to prevent, monitor and control the spread of anthelmintic resistance, including the development of viable anthelmintic vaccines