Interaction of maturation delay and nonlinear birth in population and epidemic models

 A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e⁻ d ₁ T−dN(t) governs the adult population, with the death rate in previous life stages d ₁≧0. Standard assumptions are made on B(N) so that a unique equilibrium N _e exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d ₁>0, as T increases the equilibrium N _e can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R ₀ is identified. When R ₀<1, the disease dies out; when R ₀>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay. Received: 2 November 1998 / Revised version: 26 February 1999     

Population models for diseases with no recovery

An S I epidemic model with a general shape of density-dependent mortality and incidence rate is studied. The asymptotic behaviour is global convergence to an endemic equilibrium, above a threshold, and to a disease-free equilibrium, below the threshold. The effect of vaccination is then examined.     

Time-dependent fluorescence depolarization and Brownian rotational diffusion coefficients of macromolecules

Using a distribution function formalism, we have derived relations between the time-dependent fluorescence polarization anistropy r(t) = [I‖(t) ? I?(t)]/[I‖(t) + 2I?(t)] and the Brownian rotational diffusion coefficients of macromolecules. It is shown that in the most general case of a completely asymmetric body, five exponentials appear in r(t). Reduction to previously obtained simpler expressions are presented and discussed. Experiments were performed to measure r(t) for the complex of 1-anilino-8-naphthalene sulfonate (ANS) and apomyoglobin. The results revealed a spherical geometry, with a radius of 20.8 Å. Similar experiments were performed on the ANS–apohorse radish peroxidase complex. The results were inconclusive towards the geometry of the complex, but when assumed to be spherical, the radius was estimated to be 29.6 Å, consistent with the larger molecular weight of horse radish peroxidase.     

Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission

Lyapunov functions for two-dimension SIR and SIRS compartmental epidemic models with non-linear transmission rate of a very general form f(S,I) constrained by a few biologically feasible conditions are constructed. Global properties of these models including these with vertical and horizontal transmission, are thereby established. It is proved that, under the constant population size assumption, the concavity of the function f(S,I) with respect to the number of the infective hosts I ensures the uniqueness and the global stability of the positive endemic equilibrium state. AMS Classification 92D30 (primary), 34D20 (secondary)     

Analysis of a disease transmission model in a population with varying size

An S I R S epidemiological model with vital dynamics in a population of varying size is discussed. A complete global analysis is given which uses a new result to establish the nonexistence of periodic solutions. Results are discussed in terms of three explicit threshold parameters which respectively govern the increase of the total population, the existence and stability of an endemic proportion equilibrium and the growth of the infective population. These lead to two distinct concepts of disease eradication which involve the total number of infectives and their proportion in the population.Partially supported by NSF Grant No. DMS-8703631. This work was done while this author was visiting the University of VictoriaResearch supported in part by NSERC A-8965     

Natural selection and fitness entropy in a density-regulated population

The entropy H(p_o,p*) of a population with the initial allele frequency p_o given the equilibrium polymorphic frequency p* has been proposed as a measure of natural selection. In the present paper, we have extended this concept to include a particular aspect of density-dependent selection. We compared size trajectory of a population initially at genetic equilibrium, N(t), with the size trajectories of populations not initially at p*,N(t), but which do eventually converge to a common equilibrium allele frequency and equilibrium density, N*. The following experimentally testable hyopthesis was established. The total area defined by the difference between the trajectories of N(t) and N(t) as they converge to N* is directly proportional to the fitness entropy when population size is transformed using the density-dependent fitness value. Two properties of this relationship were noted. First, it is independent of the magnitude of natural selection and, secondly, it does not depend upon the initial population density as long as the equilibrium and nonequilibrium populations have the same initial numbers. This hypothesis was evaluated with experimental data on the flour beetle Tribolium castaneum.     

The convergence in distribution of some simple epidemics

A group of n susceptible individuals exposed to a contagious disease isconsidered. It is assumed that at each point in time one or more susceptible individuals can contract the disease. The progress of this simple batch epidemic is modeled by a stochastic process X_n(t), t[0, ∞), representing the number of infectiveindividuals at time t. In this paper our analysis is restricted to simple batch epidemics with transition rates given by [α²X_n(t){n −X_n(t) +X_n(0)}]^1/2, t[0, ∞), α(0, ∞). This class of simple batch epidemics generalizes a model used and motivated by McNeil (1972) to describe simple epidemic situations. It is shown for this class of simple batch epidemics, that X_n(t), with suitable standardization, converges in distribution as n→∞ to a normal random variable for all t(0, t₀), and t₀ is evaluated.     

A model of the complex response of Staphylococcus aureus to methicillin

It is widely accepted that β-lactam antimicrobials cause cell death through a mechanism that interferes with cell wall synthesis. Later studies have also revealed that β-lactams modify the autolysis function (the natural process of self-exfoliation of the cell wall) of cells. The dynamic equilibrium between growth and autolysis is perturbed by the presence of the antimicrobial. Studies with Staphylococcus aureus to determine the minimum inhibitory concentration (MIC) have revealed complex responses to methicillin exposure. The organism exhibits four qualitatively different responses: homogeneous sensitivity, homogeneous resistance, heterogeneous resistance and the so-called ‘Eagle-effect’. A mathematical model is presented that links antimicrobial action on the molecular level with the overall response of the cell population to antimicrobial exposure. The cell population is modeled as a probability density function F(x,t) that depends on cell wall thickness x and time t. The function F(x,t) is the solution to a Fokker-Planck equation. The fixed point solutions are perturbed by the antimicrobial load and the advection of F(x,t) depends on the rates of cell wall synthesis, autolysis and the antimicrobial concentration. Solutions of the Fokker-Planck model are presented for all four qualitative responses of S. aureus to methicillin exposure.     

Photosynthesis in dynamic light: systems biology of unconventional chlorophyll fluorescence transients in Synechocystis sp. PCC 6803

Photosynthetic organisms live in a dynamic environment where light typically fluctuates around a mean level that is slowly drifting during the solar day. We show that the far-from-equilibrium photosynthesis occurring in a rapidly fluctuating light differs vastly from the stationary-flux photosynthesis attained in a constant or slowly drifting light. Photosynthetic organisms in a static or slowly drifting light can be characterized by a steady-state quantum yield of chlorophyll fluorescence emission F′ that is changing linearly with small and slow variations of the incident irradiance I+ΔI(t): F′(I+ΔI(t))≈ F_mean^′(dF)/(dI)·ΔI(t). In Synechocystis sp. PCC 6803, the linear approximation holds for an extended interval covering largely the static irradiance range experienced by the cyanobacteria in nature. The photosynthetic dynamism and, consequently, the dynamism of the chlorophyll fluorescence emission change dramatically when exposing the organism to a fluctuating irradiance. Harmonically-modulated irradiance I+ΔI · sin(2πt/T), T ≈ 1–25 s induces perpetual, far-from-equilibrium forced oscillations that are strongly non-linear, exhibiting significant hysteresis with multiple fluorescence levels corresponding to a single instantaneous level of the incident irradiance. We propose that, in nature, the far-from-equilibrium dynamic phenomena represent a significant correction to the steady-state photosynthetic activity that is typically investigated in laboratory. Analysis of the forced oscillations by the tools of systems biology suggests that the dynamism of photosynthesis observed in fluctuating light can be explained by a delayed action of regulatory agents.