Mean-square stability of a stochastic model for bacteriophage infection with time delays

We consider the stability properties of the positive equilibrium of a stochastic model for bacteriophage infection with discrete time delay. Conditions for mean-square stability of the trivial solution of the linearized system around the equilibrium are given by the construction of suitable Lyapunov functionals. The numerical simulations of the strong solutions of the arising stochastic delay differential system suggest that, even for the original non-linear model, the longer the incubation time the more the phage and bacteria populations can coexist on a stable equilibrium in a noisy environment for very long time.     

Epidemiological effects of seasonal oscillations in birth rates

Seasonal oscillations in birth rates are ubiquitous in human populations. These oscillations might play an important role in infectious disease dynamics because they induce seasonal variation in the number of susceptible individuals that enter populations. We incorporate seasonality of birth rate into the standard, deterministic susceptible-infectious-recovered (SIR) and susceptible-exposed-infectious-recovered (SEIR) epidemic models and identify parameter regions in which birth seasonality can be expected to have observable epidemiological effects. The SIR and SEIR models yield similar results if the infectious period in the SIR model is compared with the "infected period" (the sum of the latent and infectious periods) in the SEIR model. For extremely transmissible pathogens, large amplitude birth seasonality can induce resonant oscillations in disease incidence, bifurcations to stable multi-year epidemic cycles, and hysteresis. Typical childhood infectious diseases are not sufficiently transmissible for their asymptotic dynamics to be likely to exhibit such behaviour. However, we show that fold and period-doubling bifurcations generically occur within regions of parameter space where transients are phase-locked onto cycles resembling the limit cycles beyond the bifurcations, and that these phase-locking regions extend to arbitrarily small amplitude of seasonality of birth rates. Consequently, significant epidemiological effects of birth seasonality may occur in practice in the form of transient dynamics that are sustained by demographic stochasticity.     

Fixation in haploid populations exhibiting density dependence I: The non-neutral case

We extend the one-locus two allele Moran model of fixation in a haploid population to the case where the total size of the population is not fixed. The model is defined as a two-dimensional birth-and-death process for allele number. Changes in allele number occur through density-independent death events and birth events whose per capita rate decreases linearly with the total population density. Uniquely for models of this type, the latter is determined by these same birth-and-death events. This provides a framework for investigating both the effects of fluctuation in total population number through demographic stochasticity, and deterministic density-dependent changes in mean density, on allele fixation. We analyze this model using a combination of asymptotic analytic approximations supported by numerics. We find that for advantageous mutants demographic stochasticity of the resident population does not affect the fixation probability, but that deterministic changes in total density do. In contrast, for deleterious mutants, the fixation probability increases with increasing resident population fluctuation size, but is relatively insensitive to initial density. These phenomena cannot be described by simply using a harmonic mean effective population size.     

Stochastic modeling of cellular colonies with quiescence: an application to drug resistance in cancer

Several cancers are thought to be driven by cells with stem cell like properties. An important characteristic of stem cells, which also applies to primitive tumor cells, is the ability to undergo quiescence, where cells can temporarily stop the cell cycle. Cellular quiescence can affect the kinetics of tumor growth, and the susceptibility of the cells to therapy. To study how quiescence affects treatment, we formulate a stochastic birth-death process with quiescence, on a combinatorial cellular mutation network, and consider the pre-treatment (growth) and treatment (decay) regimes. We find that, in the absence of mutations, treatment (if sufficiently strong) will proceed as a biphasic decline with the first (faster) phase driven by the elimination of the cycling cells and the second (slower) phase limited by the process of cell awakening. Other regimes are possible for weaker treatments. We also describe how the process of mutant generation is influenced by quiescence. Interestingly, for single-drug treatments, the probability to have resistance at start of treatment is independent of quiescence. For two or more drugs, the probability to have generated resistant mutants before treatment grows with quiescence. Finally, we study the influence of quiescence on the treatment phase. Starting from a given composition of mutants, the chances of treatment success are not influenced by the presence of quiescence.     

A quantitative model for linking Na+/Ca2+ exchanger to SERCA during refilling of the sarcoplasmic reticulum to sustain [Ca2+] oscillations in vascular smooth muscle

We have developed a quantitative model for the creation of cytoplasmic Ca²⁺ gradients near the inner surface of the plasma membrane (PM). In particular we simulated the refilling of the sarcoplasmic reticulum (SR) via PM–SR junctions during asynchronous [Ca²⁺]_i oscillations in smooth muscle cells of the rabbit inferior vena cava. We have combined confocal microscopy data on the [Ca²⁺]_i oscillations, force transduction data from cell contraction studies and electron microscopic images to build a basis for computational simulations that model the transport of calcium ions from Na⁺/Ca²⁺ exchangers (NCX) on the PM to sarcoplasmic/endoplasmic reticulum Ca²⁺ ATPase (SERCA) pumps on the SR as a three-dimensional random walk through the PM–SR junctional cytoplasmic spaces. Electron microscopic ultrastructural images of the smooth muscle cells were elaborated with software algorithms to produce a very clear and dimensionally accurate picture of the PM–SR junctions. From this study, we conclude that it is plausible and possible for enough Ca²⁺ to pass through the PM–SR junctions to replete the SR during the regenerative Ca²⁺ release, which underlies agonist induced asynchronous Ca²⁺ oscillations in vascular smooth muscle.     

Pair approximations and the inclusion of indirect transmission: theory and application to between farm transmission of Salmonella

The spatial-temporal dynamics of farm animal diseases depend both on disease specific processes and the underlying contact network between farms. Indirect transmission via free-living bacteria in the environment is an important transmission route and contributes significantly to the dynamics. The pair-wise model has been developed to include both direct transmission and indirect transmission via free stages. The model is compared with stochastic simulations of epidemics on contact networks. The network framework is applied to the investigation of the epidemiological dynamics of between-herd transmission of Salmonella spp. The main results help to explain differences in observed epidemiological patterns and to identify possible causes for different strains of Salmonella developing so much variation in their infection dynamics in UK dairy herds. Numerical results show that shorter infectious period, more persistent immune response and more rapid removal of faeces result in a lower prevalence of infection and a greater tendency for (damped) oscillation. A possible control strategy is consequently suggested. Furthermore, the effect of network structure on long-term dynamics is examined.     

A stochastic analysis of actin polymerization in the presence of twinfilin and gelsolin

We develop an efficient stochastic simulation algorithm for analyzing actin filament growth and decay in the presence of various actin-binding proteins. The evolution of nucleotide profiles of filaments can be tracked and the resulting feedback to actin-binding proteins is incorporated. The computational efficiency of the new method enables us to focus on experimentally realistic problems, and as one example we use it to analyze the experimental data of Helfer et al. [(2006). Mammalian twinfilin sequesters ADP-G-actin and caps filament barbed ends: implications in motility. EMBO J. 25, 1184-1195] on the capping and G-actin sequestering activity of twinfilin. We show that the binding specificity of twinfilin for ADP-G-actin is crucial for the observed biphasic evolution of the filament length distribution in the presence of twinfilin, and we demonstrate that twinfilin can be an essential part of the molecular machinery for regulating filament lengths after a short burst of polymerization. Significantly, our simulations indicate that the pyrenyl-actin fluorescence experiments would fail to report the emergence of large filaments under certain experimental conditions.     

Harvesting in a random environment: Itô or Stratonovich calculus?

We extend to harvesting stochastic differential equation (SDE) models in a random environment our previous work on models without harvesting concerning the resolution of the It?-Stratonovich controversy. The resolution is obtained for the very general class of models dN/dt=N (r(N)-h(N)+sigmaepsilon(t)), where N=N(t) is the population size at time t, r(N) is the (density-dependent) "mean" per capita growth rate, h(N) is the (density-dependent) harvesting effort, epsilon(t) is a standard white noise (representing environmental random fluctuations), and sigma is a noise intensity parameter. It? and Stratonovich calculus in the resolution of SDEs apparently give different qualitative and quantitative results, leading to controversy on which calculus is more appropriate and creating an obstacle on the use of this modeling approach. We show that the apparent difference between the two calculi is due to a semantic confusion based on the fallacious assumption that we are working with the same type of mean rates. After clearing the confusion, the two calculi yield exactly the same results and we obtain important common conditions for extinction and for existence of a stationary density. The resolution of the controversy is intertwined with and sheds light on the estimation issues.     

Stochastic von Bertalanffy models, with applications to fish recruitment

We consider three individual-based models describing growth in stochastic environments. Stochastic differential equations (SDEs) with identical von Bertalanffy deterministic parts are formulated, with a stochastic term which decreases, remains constant, or increases with organism size, respectively. Probability density functions for hitting times are evaluated in the context of fish growth and mortality. Solving the hitting time problem analytically or numerically shows that stochasticity can have a large positive impact on fish recruitment probability. It is also demonstrated that the observed mean growth rate of surviving individuals always exceeds the mean population growth rate, which itself exceeds the growth rate of the equivalent deterministic model. The consequences of these results in more general biological situations are discussed.