Evolutionary dynamics of cancer cell populations under immune selection pressure and optimal control of chemotherapy

Increasing experimental evidence suggests that epigenetic and microenvironmental factors play a key role in cancer progression. In this respect, it is now generally recognized that the immune system can act as an additional selective pressure, which modulates tumor development and leads, through cancer immunoediting, to the selection for resistance to immune effector mechanisms. This may have serious implications for the design of effective anti-cancer protocols. Motivated by these considerations, we present a mathematical model for the dynamics of cancer and immune cells under the effects of chemotherapy and immunity-boosters. Tumor cells are modeled as a population structured by a continuous phenotypic trait, that is related to the level of resistance to receptor-induced cell death triggered by effector lymphocytes. The level of resistance can vary over time due to the effects of epigenetic modifications. In the asymptotic regime of small epimutations, we highlight the ability of the model to reproduce cancer immunoediting. In an optimal control framework, we tackle the problem of designing effective anti-cancer protocols. The results obtained suggest that chemotherapeutic drugs characterized by high cytotoxic effects can be useful for treating tumors of large size. On the other hand, less cytotoxic chemotherapy in combination with immunity-boosters can be effective against tumors of smaller size. Taken together, these results support the development of therapeutic protocols relying on combinations of less cytotoxic agents and immune-boosters to fight cancer in the early stages. © EDP Sciences, 2014

Coexistence and optimal control problems for a degenerate predator-prey model

In this paper we present a predator-prey mathematical model for two biological populations which dislike crowding. The model consists of a system of two degenerate parabolic equations with nonlocal terms and drifts. We provide conditions on the system ensuring the periodic coexistence, namely the existence of two non-trivial non-negative periodic solutions representing the densities of the two populations. We assume that the predator species is harvested if its density exceeds a given threshold. A minimization problem for a cost functional associated with this process and with some other significant parameters of the model is also considered. \ua9 2010 Elsevier Inc

On some problems concerning the nonlinear infiltration in unsaturated media

The paper deals with the mathematical treatment of two specific models related to water infiltration in soils. The mathematical models consist of Richard’s equation with appropriate boundary and initial conditions. The hydraulic parameters (diffusivity, hydraulic conductivity, water capacity) that represent the coefficients of this equation are nonlinear functions. Depending on the situation studied, particularities that may arise are represented by the fact that the water capacity vanishes at the saturation value, implying that the equation degenerates at the interface between unsaturated-saturated flow and diffusivity blows up at the moisture saturation value in the unsaturated model. The paper develops a theory concerning the existence of the solution of each model apart

Control Approach to an Ill-Posed Variational Inequality

We are concerned with the proof of a generalized solution to an ill-posed variational inequality. This is determined as a solution to an appropriate minimization problem involving a nonconvex functional, treated by an optimal control technique

Analysis of the time behaviour of a diffusive transport in a stratified medium

The paper presents a study of the diffusive transport of passive solute plumes in a two-dimensional non-homogeneous depth stratified flow domain. All the properties of the process are expressed by depth dependent deterministic functions. The method of moments, combined with the method of Green functions are chosen to determine the relevant characteristics of the flow (mass, center of mass, variance, etc.) used to describe the behaviour of the transient motion. General relationships for the n-order concentration moments are proved. Further, it is derived that the transient motion defined by time-dependent parameters tends asymptotically at large time to a stable regime whose characteristics are determined, Consequently, under certain hypotheses, an equivalence between the mean original process and a Fickian diffusive transport at large time may be established. The time required by the process to reach its asymptotic behaviour is also calculated