Impulsive Fire Disturbance in a Savanna Model: Tree-Grass Coexistence States, Multiple Stable System States, and Resilience.

Savanna ecosystems are shaped by the frequency and intensity of regular fires. We model savannas via an ordinary differential equation (ODE) encoding a one-sided inhibitory Lotka-Volterra interaction between trees and grass. By applying fire as a discrete disturbance, we create an impulsive dynamical system that allows us to identify the impact of variation in fire frequency and intensity. The model exhibits three different bistability regimes: between savanna and grassland; two savanna states; and savanna and woodland. The impulsive model reveals rich bifurcation structures in response to changes in fire intensity and frequency-structures that are largely invisible to analogous ODE models with continuous fire. In addition, by using the amount of grass as an example of a socially valued function of the system state, we examine the resilience of the social value to different disturbance regimes. We find that large transitions ("tipping") in the valued quantity can be triggered by small changes in disturbance regime.

Rethinking the definition of rate-induced tipping.

The current definition of rate-induced tipping is tied to the idea of a pullback attractor limiting in forward and backward time to a stable quasi-static equilibrium. Here, we propose a new definition that encompasses the standard definition in the literature for certain scalar systems and includes previously excluded N-dimensional systems that exhibit rate-dependent critical transitions.

Modeling seasonal influenza outbreak in a closed college campus: impact of pre-season vaccination, in-season vaccination and holidays/breaks.

BACKGROUND: College and university students experience substantial morbidity from influenza and influenza-like illness, and they can benefit substantially from vaccination. Public health authorities encourage vaccination not only before the influenza season but also into and even throughout the influenza season. We conducted the present study to assess the impact of various vaccination strategies including delayed (i.e., in-season) vaccination on influenza outbreaks on a college campus. METHODS/FINDINGS: We used a Susceptible --> Infected --> Recovered (SIR) framework for our mathematical models to simulate influenza epidemics in a closed, college campus. We included both students and faculty/staff in the model and derived values for the model parameters from the published literature. The values for key model parameters were varied to assess the impact on the outbreak of various pre-season and delayed vaccination rates; one-way sensitivity analyses were conducted to test the sensitivity of the model outputs to changes in selected parameter values. In the base case, with a pre-season vaccination rate of 20%, no delayed vaccination, and 1 student index case, the total attack rate (total percent infected, TAR) was 45%. With higher pre-season vaccination rates TARs were lower. Even if vaccinations were given 30 days after outbreak onset, TARs were still lower than the TAR of 69% in the absence of vaccination. Varying the proportions of vaccinations given pre-season versus delayed until after the onset of the outbreak gave intermediate TAR values. Base case outputs were sensitive to changes in infectious contact rates and infectious periods and a holiday/break schedule. CONCLUSION: Delayed vaccination and holidays/breaks can be important adjunctive measures to complement traditional pre-season influenza vaccination for controlling and preventing influenza in a closed college campus.