Global dynamics of discrete mathematical models of tuberculosis.

In this paper, we develop discrete models of Tuberculosis (TB). This includes SEI endogenous and exogenous models without treatment. These models are then extended to a SEIT model with treatment. We develop two types of net reproduction numbers, one is the traditional R0 which is based on the disease-free equilibrium, and a new net reproduction number R0(E∗) based on the endemic equilibrium. It is shown that the disease-free equilibrium is globally asymptotically stable if R0≤ 1 and unstable if R0>1. Moreover, the endemic equilibrium is locally asymptotically stable if R0(E∗)<1

A continuous-time mathematical model and discrete approximations for the aggregation of ß-Amyloid.

Alzheimer's disease is a degenerative disorder characterized by the loss of synapses and neurons from the brain, as well as the accumulation of amyloid-based neuritic plaques. While it remains a matter of contention whether ß-amyloid causes the neurodegeneration, ß-amyloid aggregation is associated with the disease progression. Therefore, gaining a clearer understanding of this aggregation may help to better understand the disease. We develop a continuous-time model for ß-amyloid aggregation using concepts from chemical kinetics and population dynamics. We show the model conserves mass and establish conditions for the existence and stability of equilibria. We also develop two discrete-time approximations to the model that are dynamically consistent. We show numerically that the continuous-time model produces sigmoidal growth, while the discrete-time approximations may exhibit oscillatory dynamics. Finally, sensitivity analysis reveals that aggregate concentration is most sensitive to parameters involved in monomer production and nucleation, suggesting the need for good estimates of such parameters.

Discrete evolutionary population models: a new approach.

In this paper, we apply a new approach to a special class of discrete time evolution models and establish a solid mathematical foundation to analyse them. We propose new single and multi-species evolutionary competition models using the evolutionary game theory that require a more advanced mathematical theory to handle effectively. A key feature of this new approach is to consider the discrete models as non-autonomous difference equations. Using the powerful tools and results developed in our recent work [E. D'Aniello and S. Elaydi, The structure of ω-limit sets of asymptotically non-autonomous discrete dynamical systems, Discr. Contin. Dyn. Series B. 2019 (to appear).], we embed the non-autonomous difference equations in an autonomous discrete dynamical systems in a higher dimension space, which is the product space of the phase space and the space of the functions defining the non-autonomous system. Our current approach applies to two scenarios. In the first scenario, we assume that the trait equations are decoupled from the equations of the populations. This requires specialized biological and ecological assumptions which we clearly state. In the second scenario, we do not assume decoupling, but rather we assume that the dynamics of the trait is known, such as approaching a positive stable equilibrium point which may apply to a much broader evolutionary dynamics.

Effects of Cholesterol in Stress-Related Neuronal Death-A Statistical Analysis Perspective.

The association between plasma cholesterol levels and the development of dementia continues to be an important topic of discussion in the scientific community, while the results in the literature vary significantly. We study the effect of reducing oxidized neuronal cholesterol on the lipid raft structure of plasma membrane. The levels of plasma membrane cholesterol were reduced by treating the intact cells with methyl-ß-cyclodextrin (MßCD). The relationship between the cell viability with varying levels of MßCD was then examined. The viability curves are well described by a modified form of the empirical Gompertz law of mortality. A detailed statistical analysis is performed on the fitting results, showing that increasing MßCD concentration has a minor, rather than significant, effect on the cellular viability. In particular, the dependence of viability on MßCD concentration was found to be characterized by a ~25% increase per 1 µM of MßCD concentration.

A discrete mathematical model for the aggregation of ß-Amyloid.

Dementia associated with the Alzheimer's disease is thought to be correlated with the conversion of the ß - Amyloid (Aß) peptides from soluble monomers to aggregated oligomers and insoluble fibrils. We present a discrete-time mathematical model for the aggregation of Aß monomers into oligomers using concepts from chemical kinetics and population dynamics. Conditions for the stability and instability of the equilibria of the model are established. A formula for the number of monomers that is required for producing oligomers is also given. This may provide compound designers a mechanism to inhibit the Aß aggregation.

Hierarchical competition models with the Allee effect II: the case of immigration.

This is part II of an earlier paper that dealt with hierarchical models with the Allee effect but with no immigration. In this paper, we greatly simplify the proofs in part I and provide a proof of the global dynamics of the non-hyperbolic cases that were previously conjectured. Then, we show how immigration to one of the species or to both would, drastically, change the dynamics of the system. It is shown that if the level of immigration to one or to both species is above a specified level, then there will be no extinction region where both species go to extinction.

Hierarchical competition models with Allee effects.

We consider a two-species hierarchical competition model with a strong Allee effect. The Allee effect is assumed to be caused by predator saturation. Moreover, we assume that there is a 'silverback' species x that gets first choice of the resources and where growth is limited by its own intraspecific competition, while the second 'inferior' species y gets whatever is left. Both species x and y are assumed to have the property of strong Allee effect. In this paper we determine the impact of the presence of the Allee effect on the global dynamics of both species.

Stability and invariant manifolds of a generalized Beddington host-parasitoid model.

We will investigate the stability and invariant manifolds of a new discrete host-parasitoid model. It is a generalization of the Beddington-Nicholson-Bailey model. Our study establishes analytically, for the first time, the stability of the coexistence fixed point.

Population models with Allee effect: a new model.

In this paper, we develop several population models with Allee effects. We start by defining the Allee effect as a phenomenon in which individual fitness increases with increasing density. Based on this biological assumption, we develop several fitness functions that produce corresponding models with Allee effects. In particular, a rational fitness function yields a new mathematical model, which is the focus of our study. Then we study the dynamics of 2-periodic systems with Allee effects and show the existence of an asymptotically stable 2-periodic carrying capacity.

Periodic difference equations, population biology and the Cushing-Henson conjectures.

We show that for a k-periodic difference equation, if a periodic orbit of period r is globally asymptotically stable (GAS), then r must be a divisor of k. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has a GAS periodic orbit with minimum period r. Our method uses the technique of skew-product dynamical systems. Our methods are then applied to prove two conjectures of Cushing and Henson concerning a non-autonomous Beverton-Holt equation which arises in the study of the response of a population to a periodically fluctuating environmental force such as seasonal fluctuations in carrying capacity or demographic parameters like birth or death rates. We give an equality linking the average population with the growth rates and carrying capacities (in the 2-periodic case) which shows that out-of-phase oscillations in these quantities always have a deleterious effect on the average population. We give an example where in-phase oscillations cause the opposite to occur.