David M. Chan

An in silico approach to the analysis of acute wound healing

The complex interactions that characterize acute wound healing have stymied the development of effective therapeutic modalities. The use of computational models holds the promise to improve our basic approach to understanding the process. By modifying an existing ordinary differential equation model of systemic inflammation to simulate local wound healing, we expect to improve the understanding of the underlying complexities of wound healing and thus allow for the development of novel, targeted therapeutic strategies. The modifications in this local acute wound healing model include: evolution from a systemic model to a local model, the incorporation of fibroblast activity, and the effects of tissue oxygenation. Using these modifications we are able to simulate impaired wound healing in hypoxic wounds with varying levels of contamination. Possible therapeutic targets, such as fibroblast death rate and rate of fibroblast recruitment, have been identified by computational analysis. This model is a step toward constructing an integrative systems biology model of human wound healing.

Asymptotic stability for difference equations with decreasing arguments

We consider general, higher order difference equations of type in which the function f is non-increasing in each coordinate. We obtain sufficient conditions for the asymptotic stability of a unique fixed point relative to an invariant interval. We also discuss various applications of our main results.

A mathematical model for antimalarial drug resistance

We formulate and analyze a mathematical model for malaria with treatment and the well-known three levels of resistance in humans. The model incorporates both sensitive and resistant strains of the parasites. Analytical results reveal that the model exhibits the phenomenon of backward bifurcation (co-existence of a stable disease-free equilibrium with a stable endemic equilibrium), an epidemiological situation where although necessary, having the basic reproduction number less than unity, it is not sufficient for disease elimination. Through quantitative analysis, we show the effects of varying treatment levels in a high transmission area with different levels of resistance. Increasing treatment has limited benefits in a population with resistant strains, especially in high transmission settings. Thus, in a cost-benefit analysis, the rate of treatment and percentage to be treated become difficult questions to address.

Probabilities of extinction, weak extinction permanence, and mutual exclusion in discrete, competitive, Lotka-Volterra systems

Abstract The probabilities of extinction, weak extinction, permanence, and mutual exclusion are calculated for models with up to five species by examining one million randomly chosen discrete, competitive, LotkarVolterra systems. The probability of permanence drops off very rapidly with the increase in the number of species. It drops to less than 1% with five species. The probability that at least one species will die out increases with the number of species. It reaches 95% with five species. When a group of species weakly dominates another species, the dominated species goes extinct. The probability that at least one species is weakly dominated is close to 50%. Mutual exclusion happens between 10% and 20% of the time when there are at least three species.

Convergence Results on a Second-Order Rational Difference Equation with Quadratic Terms

We investigate the global behavior of the second-order difference equation , where initial conditions and all coefficients are positive. We find conditions on under which the even and odd subsequences of a positive solution converge, one to zero and the other to a nonnegative number; as well as conditions where one of the subsequences diverges to infinity and the other either converges to a positive number or diverges to infinity. We also find initial conditions where the solution monotonically converges to zero and where it diverges to infinity.

Probabilities of extinction, weak extinction,permanence, and mutual exclusion in discrete, competitive, Lotka-Volterra systems that involve invading species

The probabilities of various biological asymptotic dynamics are computed for a stable system that is invaded by another competing species. The asymptotic behaviors studied include extinction, weak extinction, permanence, and mutual exclusion. The model used is a discrete Lotka-Volterra system that models species that compete for the same resources. Among the results found are that the chance of permanence occurring in the invaded system is significantly higher than the probability of permanence in a purely random system, and that multiple extinctions that include the invading species and one of the original species are impossible.

Management of invasive Allee species

Abstract In this study, we use a discrete, two-patch population model of an Allee species to examine different methods in managing invasions. We first analytically examine the model to show the presence of the strong Allee effect, and then we numerically explore the model to test the effectiveness of different management strategies. As expected invasion is facilitated by lower Allee thresholds, greater carrying capacities and greater proportions of dispersers. These effects are interacting, however, and moderated by population growth rate. Using the gypsy moth as an example species, we demonstrate that the effectiveness of different invasion management strategies is context-dependent, combining complementary methods may be preferable, and the preferred strategy may differ geographically. Specifically, we find methods for restricting movement to be more effective in areas of contiguous habitat and high Allee thresholds, where methods involving mating disruptions and raising Allee thresholds are more effective in areas of high habitat fragmentation.

Complex temporal patterns of spontaneous initiation and termination of reentry in a loop of cardiac tissue

A two-component model is developed consisting of a discrete loop of cardiac cells that circulates action potentials as well as a pacing mechanism. Physiological properties of cells such as restitutions of refractoriness and of conduction velocity are given via experimentally measured functions. The dynamics of circulating pulses and the pacers action are regulated by two threshold relations. Patterns of spontaneous initiations and terminations of reentry (SITR) generated by this system are studied through numerical simulations and analytical observations. These patterns can be regular or irregular; causes of irregularities are identified as the threshold bistability (T-bistability) of reentrant circulation and in some cases, also phase-resetting interactions with the pacer.

A Proposal for an Application of a Max-Type Difference Equation to Epilepsy

We propose, for the sake of dialogue, that the nonautomomous reciprocal max-type difference equation,