A Possible Scenario for the Covid-19 Epidemic, Based on the SI(R) Model.

Many attempts to build epidemic models of the current Covid-19 epidemic have been made in the recent past. However, only models postulating permanent immunity have been proposed. In this paper, we propose a SI(R) model in order to forecast the evolution of the epidemic under the hypothesis of not permanent immunity. This model offers an analytical solution to the problem of finding possible steady states, providing the following equilibrium values: Susceptible about 17%, Recovered (including deceased and healed) ranging from 79 to 81%, and Infected ranging from 2 to 4%. However, it is crucial to consider that the results concerning the recovered, which at first glance are particularly impressive, include the huge proportion of asymptomatic subjects. On the basis of these considerations, we analyse the situation in the province of Pesaro-Urbino, one of the main outbreaks of the epidemic in Italy.

Mean-square stability of a stochastic model for bacteriophage infection with time delays.

We consider the stability properties of the positive equilibrium of a stochastic model for bacteriophage infection with discrete time delay. Conditions for mean-square stability of the trivial solution of the linearized system around the equilibrium are given by the construction of suitable Lyapunov functionals. The numerical simulations of the strong solutions of the arising stochastic delay differential system suggest that, even for the original non-linear model, the longer the incubation time the more the phage and bacteria populations can coexist on a stable equilibrium in a noisy environment for very long time.

Numerical simulation of a Campbell-like stochastic delay model for bacteriophage infection.

In this work, we consider a delay differential equations model for bacteriophage infection and discuss the robustness of the positive equilibrium with respect to stochastic perturbations of the environment using two different approaches. First, we provide analytical estimates of the population intensities of fluctuations by Fourier transform methods. Next, we simulate the strong solutions of the arising stochastic delay differential equations by numerical methods of order 1. Extensive numerical experiments suggest that a noisy environment for the bacteria population is much more destabilizing on the concentrations at the equilibrium point than a noisy environment for the phage.

Stochastic modelling of PTEN regulation in brain tumors: A model for glioblastoma multiforme.

This work is the outcome of the partnership between the mathematical group of Department DISBEF and the biochemical group of Department DISB of the University of Urbino "Carlo Bo" in order to better understand some crucial aspects of brain cancer oncogenesis. Throughout our collaboration we discovered that biochemists are mainly attracted to the instantaneous behaviour of the whole cell, while mathematicians are mostly interested in the evolution along time of small and different parts of it. This collaboration has thus been very challenging. Starting from [23,24,25], we introduce a competitive stochastic model for post-transcriptional regulation of PTEN, including interactions with the miRNA and concurrent genes. Our model also covers protein formation and the backward mechanism going from the protein back to the miRNA. The numerical simulations show that the model reproduces the expected dynamics of normal glial cells. Moreover, the introduction of translational and transcriptional delays offers some interesting insights for the PTEN low expression as observed in brain tumor cells.