Serena Spina

On some time non-homogeneous queueing systems with catastrophes

Computations for non-homogeneous birth-and-death processes with catastrophes.Applications to queueing and population dynamics problems.Effect of the periodicity of the rates on the performance of systems with catastrophes. Non-stationary queueing systems subject to catastrophes occurring with time varying intensity are considered. The effect of a catastrophe is to make the queue instantly empty. The transition probabilities, the related moments and the first visit time density to zero state are analyzed. Particular attention is dedicated to queueing systems in the presence of catastrophes with periodic intensity function. Various applications are provided, including the non-stationary birth-death process with immigration, the queueing systems M ( t ) / M ( t ) / 1 and M ( t ) / M ( t ) / ∞ .

On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model

An Ornstein-Uhlenbeck diffusion process is considered as a model for the membrane potential activity of a single neuron. We assume that the neuron is subject to a sequence of inhibitory and excitatory post-synaptic potentials that occur with time-dependent rates. The resulting process is characterized by time-dependent drift. For this model, we construct the return process describing the membrane potential. It is a non homogeneous Ornstein-Uhlenbeck process with jumps on which the effect of random refractoriness is introduced. An asymptotic analysis of the process modeling the number of firings and the distribution of interspike intervals is performed under the assumption of exponential distribution for the firing time. Some numerical evaluations are performed to provide quantitative information on the role of the parameters.

Estimating a non-homogeneous Gompertz process with jumps as model of tumor dynamics

A non-homogeneous stochastic model based on a Gompertz-type diffusion process with jumps is proposed to describe the evolution of a solid tumor subject to an intermittent therapeutic program. Each therapeutic application, represented by a jump in the process, instantly reduces the tumor size to a fixed value and, simultaneously, increases the growth rate of the model to represent the toxicity of the therapy. This effect is described by introducing a time-dependent function in the drift of the process. The resulting model is a combination of several non-homogeneous diffusion processes characterized by different drifts, whose transition probability density function and main characteristics are studied. The study of the model is performed by distinguishing whether the therapeutic instances are fixed in advance or guided by a strategy based on the mean of the first-passage-time through a control threshold. Simulation studies are carried out for different choices of the parameters and time-dependent functions involved.

Analysis of a growth model inspired by Gompertz and Korf laws, and an analogous birth-death process

We propose a new deterministic growth model which captures certain features of both the Gompertz and Korf laws. We investigate its main properties, with special attention to the correction factor, the relative growth rate, the inflection point, the maximum specific growth rate, the lag time and the threshold crossing problem. Some data analytic examples and their performance are also considered. Furthermore, we study a stochastic counterpart of the proposed model, that is a linear time-inhomogeneous birth-death process whose mean behaves as the deterministic one. We obtain the transition probabilities, the moments and the population ultimate extinction probability for this process. We finally treat the special case of a simple birth process, which better mimics the proposed growth model.

A Stochastic Model of Cancer Growth Subject to an Intermittent Treatment with Combined Effects: Reduction in Tumor Size and Rise in Growth Rate

A model of cancer growth based on the Gompertz stochastic process with jumps is proposed to analyze the effect of a therapeutic program that provides intermittent suppression of cancer cells. In this context, a jump represents an application of the therapy that shifts the cancer mass to a return state and it produces an increase in the growth rate of the cancer cells. For the resulting process, consisting in a combination of different Gompertz processes characterized by different growth parameters, the first passage time problem is considered. A strategy to select the inter-jump intervals is given so that the first passage time of the process through a constant boundary is as large as possible and the cancer size remains under this control threshold during the treatment. A computational analysis is performed for different choices of involved parameters. Finally, an estimation of parameters based on the maximum likelihood method is provided and some simulations are performed to illustrate the validity of the proposed procedure.

A Stochastic Gompertz Model with Jumps for an Intermittent Treatment in Cancer Growth

To analyze the effect of a therapeutic program that provides intermittent suppression of cancer cells, we suppose that the Gompertz stochastic diffusion process is influenced by jumps that occur according to a probability distribution, producing instantaneous changes of the system state. In this context a jump represents an application of the therapy that leads the cancer mass to a return state randomly chosen. In particular, constant and exponential intermittence distribution are considered for different choices of the return state. We perform several numerical analyses to understand the behavior of the process for different choices of intermittence and return point distributions.

A Cancer Dynamics Model for an Intermittent Treatment Involving Reduction of Tumor Size and Rise of Growth Rate

We propose a model of tumor dynamics based on the Gompertz deterministic law influenced by jumps that occur at equidistant time instants. This model consents to study the effect of a therapeutic program that provides intermittent suppression of cancer cells. In this context a jump represents an application of the therapy that shifts the cancer mass to a fixed level and it produces a deleterious effect on the organism by increasing the growth rate of the cancer cells. The objective of the present study is to provide an efficient criterion to choose the instants in which to apply the therapy by maximizing the time in which the cancer mass is under a fixed control threshold.

A Note on Diffusion Processes with Jumps

We focus on stochastic diffusion processes with jumps occurring at random times. After each jump the process is reset to a fixed state from which it restarts with a different dynamics. We analyze the transition probability density function, its moments and the first passage time density. The obtained results are used to study the lognormal diffusion process with jumps which is of interest in the applications.