Inference on an heteroscedastic Gompertz tumor growth model.

We consider a non homogeneous Gompertz diffusion process whose parameters are modified by generally time-dependent exogenous factors included in the infinitesimal moments. The proposed model is able to describe tumor dynamics under the effect of anti-proliferative and/or cell death-induced therapies. We assume that such therapies can modify also the infinitesimal variance of the diffusion process. An estimation procedure, based on a control group and two treated groups, is proposed to infer the model by estimating the constant parameters and the time-dependent terms. Moreover, several concatenated hypothesis tests are considered in order to confirm or reject the need to include time-dependent functions in the infinitesimal moments. Simulations are provided to evaluate the efficiency of the suggested procedures and to validate the testing hypothesis. Finally, an application to real data is considered.

Modeling Environmental Influences in the Psyllaephagus bliteus (Hymenoptera: Encyrtidae)-Glycaspis brimblecombei (Hemiptera: Aphalaridae) Parasitoid-Host System.

Glycaspis brimblecombei Moore (Hemiptera: Aphalaridae) is an invasive psyllid introduced into the Mediterranean area, where it affects several species of Eucalyptus. Psyllaephagus bliteus Riek (Hymenoptera: Encyrtidae) is a specialized parasitoid of this psyllid that was accidentally introduced into Italy in 2011. We developed a model of this host-parasitoid system that accounts for the influence of environmental conditions on the G. brimblecombei population dynamics and P. bliteus parasitism rates in the natural ecosystem. The Lotka-Volterra-based model predicts non-constant host growth and parasitoid mortality rates in association with variation in environmental conditions. The model was tested by analyzing sampling data collected in Naples in 2011 (before the parasitoid was present) and defining several environmental patterns, termed Temperature-Rain or T-R patterns, which correspond to the host growth rate. A mean value of the host growth rate was assigned to each T-R pattern, as well as a variation of the parasitoid mortality rate based on temperature thresholds. The proposed model was applied in simulation tests related to T-R patterns carried out with a data series sampled between June 2014 and July 2015 in five Italian sites located in Campania, Lazio, Sicily, and Sardinia regions. The simulation results showed that the proposed model provides an accurate approximation of population trends, although oscillation details may not be apparent. Results predict a 64% reduction in G. brimblecombei population density owing to P. bliteus parasitoid activity. Our results are discussed with respect to features of the host-parasitoid interaction that could be exploited in future biological control programs.

A stochastic model in tumor growth.

A stochastic model of solid tumor growth based on deterministic Gompertz law is presented. Tumor cells evolution is described by a one-dimensional diffusion process limited by two absorbing boundaries representing healing threshold and patient death (carrying capacity), respectively. Via a numerical approach the first exit time problem is analysed for the process inside the region restricted by the boundaries. The proposed model is also implemented to simulate the effects of a time-dependent therapy. Finally, some numerical results are obtained for the specific case of a parathyroid tumor.

A neuronal modeling paradigm in the presence of refractoriness.

A mathematical characterization of the membrane potential as an instantaneous return process in the presence of refractoriness is investigated for diffusion models of single neuron's activity, assuming that the firing threshold acts as an elastic barrier. Steady-state probability densities and asymptotic moments of the neuronal membrane potential are explicitly obtained in a form that is suitable for quantitative evaluations. For the Ornstein-Uhlenbeck (OU) and Feller neuronal models, closed form expression are obtained for asymptotic mean and variance of the neuronal membrane potential and an analysis of the different features exhibited by the above mentioned models is performed.

Stochastic population models with interacting species.

A stochastic model concerning the evolution of a multi-species population is presented assuming species competition for a habitat. The model takes into account colonization, death and replacement for all individuals. Two cases are treated: (i) colonizations follow the hierarchic rule by which species of lower rank are always outcompeted by those of higher rank and (ii) there are no privileged species. In both cases, under suitable assumptions, a thorough description of the evolution of the population is obtained. The two models are finally compared and the corresponding evolutionary behaviors of the populations are discussed.

Single neuron's activity: on certain problems of modeling and interpretation.

With reference to the Ornstein-Uhlenbeck model for single neuron activity, computational results and theoretical arguments are provided to discuss the accuracy and the appropriateness of analytical approximations to first-passage-time densities and its moments. A gamma approximation is initially discussed, use of which is successively made to construct a probability density of a new form that appears to be particularly suitable to approximate the as yet unknown firing probability density function.

Diffusion approximation and first-passage-time problem for a model neuron. III. A birth-and-death process approach.

A stochastic model for single neuron's activity is constructed as the continuous limit of a birth-and-death process in the presence of a reversal hyperpolarization potential. The resulting process is a one dimensional diffusion with linear drift and infinitesimal variance, somewhat different from that proposed by Lánský and Lánská in a previous paper. A detailed study is performed for both the discrete process and its continuous approximation. In particular, the neuronal firing time problem is discussed and the moments of the firing time are explicitly obtained. Use of a new computation method is then made to obtain the firing p.d.f. The behaviour of mean, variance and coefficient of variation of the firing time and of its p.d.f. is analysed to pinpoint the role played by the parameters of the model. A mathematical description of the return process for this neuronal diffusion model is finally provided to obtain closed form expressions for the asymptotic moments and steady state p.d.f. of the neuron's membrane potential.