Scientific Articles in Peer-Reviewed International Journals

Authors | C. Bianca, B. Carbonaro, M. Menale |

Title | On the Cauchy problem of vectorial thermostatted kinetic frameworks |

Journal | Symmetry 12 (2020), 517 |

Abstract
| This paper is devoted to the derivation and mathematical analysis of new thermostatted kinetic theory frameworks for the modeling of nonequilibrium complex systems composed by particles whose microscopic state includes a vectorial state variable. The mathematical analysis refers to the global existence and uniqueness of the solution of the related Cauchy problem. Specifically, the paper is divided in two parts. In the first part the thermostatted framework with a continuous vectorial variable is proposed and analyzed. The framework consists of a system of partial integro-differential equations with quadratic type nonlinearities. In the second part the thermostatted framework with a discrete vectorial variable is investigated. Real world applications, such as social systems and crowd dynamics, and future research directions are outlined in the paper. |

Authors | M. Dalla Via, C. Bianca, I. El Abbassi, A-M. Darcherif |

Title | On the modeling of a solar, wind and fossil fuel energy source by means of the thermostatted kinetic theory |

Journal | The European Physical Journal Plus 135 (2020), 198 |

Abstract
| This paper is devoted to the modeling of a hybrid energy distribution network with storage, within the thermostatted kinetic theory framework. The network consists of a non-renewable energy source and a renewable energy source. The energy storage is modeled by the introduction of the external force field coupled to the thermostat term. The activation parameters of the energy sources are assumed time-dependent in order to mimic the timedependent efficiency of different specific energy sources. In particular a solar energy source, a wind energy source and a fossil fuel energy source aremodeled.Acomputational analysis is performed to showthe effects of the intermittent activation on the plan of quality improvement of the energy provided to the customers and on the construction of the energy storage. Discussions and future research perspectives are proposed in the last section of the paper. |

Authors | C. Bianca, M. Menale |

Title | Mathematical analysis of a thermostatted equation with a discrete real activity variable |

Journal | Mathematics 8 (2020), 57 |

Abstract
| This paper deals with the mathematical analysis of a thermostatted kinetic theory equation. Specifically, the assumption on the domain of the activity variable is relaxed allowing for the discrete activity to attain real values. The existence and uniqueness of the solution of the related Cauchy problem and of the related non-equilibrium stationary state are established, generalizing the existing results. |

Authors | M. Dalla Via, C. Bianca, I. El Abbassi, A-M. Darcherif |

Title | A thermostatted kinetic theory model for a hybrid multisource system with storage |

Journal | Applied Mathematical Modelling 78 (2020), 232-248 |

Abstract
| This paper deals with the modeling of a hybrid energy multisource network composed by a non-renewable energy source and a renewable energy source. The mathematical model is derived within the framework of the thermostatted kinetic theory where the external force field coupled to the thermostat term mimics the construction of the energy storage. The parameters of the mathematical model are set in order to promote the use of the renewable energy source thus improving the quality of the provided energy. A computational analysis is performed to show the emerging phenomena that the model is able to capture. Specifically the computational analysis is mainly addressed to a sensitivity analysis on the switching-source parameters and the transition-energy parameters. Moreover the construction of the energy storage is analyzed by performing a sensitivity analysis on the magnitude of the external force field. Discussions and future research perspectives are postponed to the last section of the paper. |

Authors | C. Bianca, M. Menale |

Title | The maximum-entropy-based weight function in discrete-activity-thermostatted models |

Journal | Applied Mathematics & Information Sciences 14 (2020), to appear |

Abstract
| Recently the weighted thermostatted kinetic theory framework with discrete activity has been proposed for the modeling of the weighted interactions in complex systems. This paper deals with the derivation of the weight function as a solution of an inverse problem based on the macroscopic quantities and specifically on the high-order moments. The weight function, obtained by employing the maximum entropy principle, generalizes the previous published results obtained by employing the zero-order moment. |

Authors | C. Bianca, M. Menale |

Title | On the convergence toward nonequilibrium stationary states in thermostatted kinetic models |

Journal | Mathematical Methods in the Applied Sciences 42 (2019), 6624-6634 |

Abstract
| A differential equation-based framework is suitable for themodeling of nonequilibrium complex systems if its solution is able to reach, as time goes to infinity, the related nonequilibrium steady states. The thermostatted kinetic theory framework has been recently proposed for the modeling of complex systems subjected to an external force field. The present paper is devoted to the mathematical proof of the convergence of the solutions of the thermostatted kinetic framework towards the related nonequilibrium stationary states. The proof of the main result is gained by employing the Fourier transform and distribution theory arguments. |

Authors | C. Bianca, M. Menale |

Title | On the interaction domain reconstruction in the weighted thermostatted kinetic framework |

Journal | The European Physical Journal Plus 134 (2019), 143 |

Abstract
| This paper is devoted to the modeling of out-of-equilibrium complex living systems by means of the weighted thermostatted kinetic theory framework. The weighted mathematical framework is based on the definition and interaction of different functional subsystems each of them able to express a specific strategy. The time evolution of the functional subsystems is described by nonlinear partial integro-differential equations with quadratic type nonlinearity coupled with a thermostat in order to ensure the reaching of nonequilibrium stationary states. In particular the weighted framework is based on the definition of the weighted interactions which are modeled by introducing an interaction domain. This paper focuses on the interaction domain reconstruction by employing the methods of the inverse theory and the information theory. Specifically the solution of different inverse problems based on the knowledge of global weighted measurements related to the system is investigated. An optimization problem based on the maximum entropy principle of Shannon is analyzed and the existence of the interaction domain is proven by employing fixed-point arguments. Applications to living systems, such as social systems and crowd dynamics, and further research directions are outlined in the last section of the paper. |

Authors | C. Bianca, M. Menale |

Title | Existence and uniqueness of nonequilibrium stationary solutions in discrete thermostatted models |

Journal | Commun Nonlinear Sci Numer Simulat 73 (2019), 25-34 |

Abstract
| This paper deals with the mathematical proof of the existence of the nonequilibrium sta- tionary states in complex systems subjected to external force fields. Specifically the com- plex system is modeled within the recently proposed framework of the thermostatted ki- netic theory. As the main result shows, the thermostat term allows the existence of the nonequilibrium stationary solutions, however a lower bound on the magnitude of the force field is required in order to ensure its uniqueness. The main result is gained by fixed point arguments. |

Authors | C. Bianca, C. Dogbe |

Title | Regularity of entropy solutions to a class of conservation laws |

Journal | Nonlinear Studies 26 (2019), 129-157 |

Abstract
| This paper deals with the analysis of solutions of a class of conservation laws when the flux function is a x-dependent function. Specifically we prove that, under less regular assumptions on the flux, the maximum between two entropy subsolutions is also an entropy subsolution. The proof of the main result is based on the entropy definition of the Kruzkhov theory, the Lions-Souganidis methods and the regularization method of DiPerna-Lions. |

Authors | C. Bianca, M. Menale |

Title | A convergence theorem for the nonequilibrium states in the discrete thermostatted kinetic theory |

Journal | Mathematics 7 (2019), 673 |

Abstract
| The existence and reaching of nonequilibrium stationary states are important issues that need to be taken into account in the development of mathematical modeling frameworks for far off equilibrium complex systems. The main result of this paper is the rigorous proof that the solution of the discrete thermostatted kinetic model catches the stationary solutions as time goes to infinity. The approach towards nonequilibrium stationary states is ensured by the presence of a dissipative term (thermostat) that counterbalances the action of an external force field. The main result is obtained by employing the Discrete Fourier Transform (DFT). |

Authors | L. Masurel, C. Bianca, A. Lemarchand |

Title | Thermostatted kinetic theory approach to the competition between cancer and immune system cells in an inhomogeneous system |

Journal | AIP Conference Proceedings 2131 (2019) |

Abstract
| The competition between a cancer and the immune system are modelled at cell scale in the framework of thermostatted kinetic theory. Cell activation and learning are reproduced by the increase of cell activity during interactions. The fluctuations of system activity are controlled by a thermostat which reproduces the regulation of the learning process and memory loss through cell death. An algorithm, including spatial description and inspired from the direct simulation Monte Carlo (DSMC) method, is used to simulate stochastic trajectories for cell numbers and activities. We focus on the decisive role played by the thermostat. For inefficient thermalization, the divergence of the number of cancer cells is obtained in spite of favored production of immune system cells. Conversely, when the activity fluctuations are controlled, the development of cancer is contained even for weakened immune defenses. These results may be correlated to unexpected clinical observations in the case of different cancers, such as carcinoma, lymphoma, and melanoma. |

Authors | C. Bianca, M. Dalla Via, C. Dogbe |

Title | A master equation-based framework for the modeling of pedestrian dynamics |

Journal | Mathematics in Engineering, Science and Aerospace 10 (2019), 129-142 |

Abstract
| This paper is devoted to the derivation of a master equation-based framework for the modeling of the pedestrian dynamics into a bounded domain of the plane. The framework is based on the domain decomposition into squares with length side e and on the definition of the transition rates in the admissible directions that define the mathematical operators of the master equation fulfilled by the joint probability. In particular two specific mathematical models are derived within the new framework: An isotropic model for the pedestrian dynamics in the checkout area of a supermarket where the pedestrians move towards low-density regions of the domain, and an anisotropic model describing the movement of pedestrians during an evacuation where high-density regions of the domain are reached. The macroscopic dynamics, obtained by letting e go to zero, is described by reaction-diffusion equations. |

Authors | C. Bianca, M. Menale |

Title | On the weighted interactions in the discrete thermostatted kinetic theory |

Journal | Nonlinear Studies 26 (2019), 95-108 |

Abstract
| The mathematical modeling of the interactions among the active particles of a complex system is an important issue in the discrete thermostatted kinetic theory. In particular a weight function needs to be introduced in order to differentiate the multiple interactions. This paper is devoted to the definition of a weighted framework of the discrete thermostatted kinetic theory. Specifically the weight function is assumed to be solution of an inverse problem based on the weighted global density of the system. The general existence and uniqueness of the weight function is proved by employing the maximum entropy principle of Shannon. Moreover the analytical expression of the weight function is derived. Finally application to complex systems and research perspectives are also discussed in the last section of the paper. |

Authors | C. Bianca, S. Motta |

Title | Optimal control in a mathematical model for tumor escape |

Journal | IEEE BIBM (2019), 1357-1360 |

Abstract
| This paper deals with the medical treatment of tumors which are able to escape the immune system surveillance. A mathematical model for the tumor immune system competition is generalized by introducing a therapy which helps the immune system cells to recognize the tumor cells. The mathematical model consists into a system of two ordinary differential equations modeling the time evolution of the density of the immune system cells and the density of the tumor cells, respectively, and their competition. An optimal control strategy is introduced in order to reduce the percentage of tumor cells which are able to escape the immune system action. Specifically the control represents the percentage of effect that the therapy has on the tumor escape phenomenon production. The therapy lasts for a given period; the results, however, are not interval-dependent. The objective function to be maximized is based on the benefit related to T-cell counts less the systemic cost of the therapy. The model is mathematically analyzed and the existence of the optimal control is investigated by using the Pontryagins Maximum Principle. |

Authors | C. Bianca, C. Mogno |

Title | A thermostatted kinetic theory model for event-driven pedestrian dynamics |

Journal | The European Physical Journal Plus 133 (2018), 213 |

HIGHLIGHTS | |

Abstract
| This paper is devoted to the modeling of the pedestrian dynamics by means of the thermostatted kinetic theory. Specifically the microscopic interactions among pedestrians and an external force field are modeled for simulating the evacuation of pedestrians from a metro station. The fundamentals of the stochastic game theory and the thermostatted kinetic theory are coupled for the derivation of a specific mathematical model which depicts the time evolution of the distribution of pedestrians at different exits of a metro station. The perturbation theory is employed in order to establish the stability analysis of the nonequilibrium stationary states in the case of a metro station consisting of two exits. A general sensitivity analysis on the initial conditions, the magnitude of the external force field and the number of exits is presented by means of numerical simulations which, in particular, show how the asymptotic distribution and the convergence time are affected by the presence of an external force field. The results show how, in evacuation conditions, the interaction dynamics among pedestrians can be negligible with respect to the external force. The important role of the thermostat term in allowing the reaching of the nonequilibrium stationary state is stressed out. Research perspectives are underlined at the end of paper, in particular for what concerns the derivation of frameworks that take into account the definition of local external actions and the introduction of the space and velocity dynamics. |

Authors | C. Bianca, C. Mogno |

Title | Modelling pedestrian dynamics into a metro station by thermostatted kinetic theory methods |

Journal | Mathematical and Computer Modelling of Dynamical Systems 24 (2018), 207-235 |

Abstract
| This paper deals with the modelling of pedestrian dynamics at the entry of a metro station by means of the thermostatted kinetic theory framework. Specifically, the model depicts the time evolution of the pedestrian dynamics at the turnstiles under no panic conditions. The modelling of the microscopic interactions is based on the stochastic game theory and reflects the decision dynamics of the turnstiles pursued by pedestrians. A qualitative analysis is addressed to the equilibrium solutions by means of the classical stability theory of perturbations. Numerical simulations aim at showing the emerging behaviours captured by the model. In particular the model validation is obtained by performing a sensitivity analysis on the parameters and on the initial conditions. Further refinements and research perspective, including the modelling under panic conditions, are discussed in the last section of the paper. |

Authors | L. Masurel, C. Bianca, A. Lemarchand |

Title | On the learning control effects in the cancer-immune system competition |

Journal | Physica A: Statistical Mechanics and its Applications 506 (2018), 462-475 |

Abstract
| The interactions between a tumor and the immune system are modeled at cell scale in the framework of thermostatted kinetic theory. Cell activation and learning are reproduced by the increase of cell activity during interactions. The second moment of the activity of the whole system is controlled by a thermostat which reproduces the regulation of the learning process and memory loss through cell death. An algorithm inspired from the direct simulation Monte Carlo (DSMC) method is used to simulate stochastic trajectories for the numbers of cells and to study the sensitivity of the dynamics to various parameters. The nonintuitive role played by the thermostat is pointed out. For inefficient thermalization, the divergence of the number of cancer cells is obtained in spite of favored production of immune system cells. Conversely, when the activity fluctuations are controlled, the development of cancer is contained even for weakened immune defenses. These results may be correlated to unexpected clinical observations in the case of different cancers, such as carcinoma, lymphoma, and melanoma. |

Authors | C. Bianca, C. Mogno |

Title | Qualitative analysis of a discrete thermostatted kinetic framework modeling complex adaptive systems |

Journal | Commun Nonlinear Sci Numer Simulat 54 (2018), 221-232 |

Abstract
| This paper deals with the derivation of a new discrete thermostatted kinetic framework for the modeling of complex adaptive systems subjected to external force fields (nonequilibrium system). Specifically, in order to model nonequilibrium stationary states of the system, the external force field is coupled to a dissipative term (thermostat). The well-posedness of the related Cauchy problem is investigated thus allowing the new discrete thermostatted framework to be suitable for the derivation of specific models and the related computational analysis. Applications to crowd dynamics and future research directions are also discussed within the paper. |

Authors | C. Bianca, C. Dogbe |

Title | A new criterium for the ergodicity of Hamilton-Jacobi-Bellman type equations |

Journal | Global and Stochastic Analysis 5 (2018), 67-99 |

Abstract
| This paper deals with the link among the large-time behavior of a class of fully nonlinear partial differential equations, the concept of mean ergodicity of a dynamical system and the controllability problem. Specifically Abelian-Tauberian arguments are employed to develop a theory for the analysis of the ergodic mean behavior of systems of degenerate elliptic-parabolic equations and general systems of vector fields satisfying Hormander’s condition. A new criterium for ergodicity is established which is based on an asymptotic estimation of the rate of convergence. The new criterium is employed for the asymptotic analysis of Hamilton-Jacobi-Bellman type equations. |

Authors | C. Bianca, R. Sasportas, X. Busch |

Title | A two-dimensional maximum-entropy-based model for the source reconstruction in a monitoring network |

Journal | Nonlinear Studies 25 (2018), 1-13 |

Abstract
| This paper deals with the derivation of a two-dimensional model within a new theoretical framework that has been recently proposed for the source localization into a domain where a network of detectors is arranged. Specifically the framework is based on the resolution of an inverse problem (source reconstruction) by employing the maximum Shannon entropy principle. The framework is proposed as a general paradigm for the derivation of a specific model and a sensitivity analysis on the detectors position is addressed. Further applications and research directions are also discussed into the last section of the paper. |

Authors | C. Bianca, L. Brézin |

Title | Modeling the antigen recognition by B-cell and T-cell receptors through thermostatted kinetic theory methods |

Journal | International Journal of Biomathematics 10 (2017), 1750072 |

Abstract
| The activation and the resulting response of the immune system to antigens comprise different complex processes and cells. This paper aims at modeling the processes of recognition and learning of the immune system by means of the thermostatted kinetic theory methods. Specifically, the thermostatted kinetic framework is firstly generalized for taking into account that in some processes of proliferation of the cells, the rate is also function of the degree of information exchanged amongst cells. In particular, within the new framework, a mathematical model is proposed for miming the recognition process of the immune system through the definition of interactions between the cytotoxic and humoral components of the adaptive immune system via T- and B-cells. The model validation is obtained by performing a sensitivity analysis on the parameters which depicts the main emerging phenomena and the different phases of the recognition and learning of the immune system. |

Authors | C. Bianca, C. Dogbe |

Title | On the existence and uniqueness of invariant measure for multidimensional diffusion processes |

Journal | Nonlinear Studies 24 (2017), 437-468 |

Abstract
| This paper deals with the mathematical analysis of multidimensional processes solution of a class of stochastic differential equations. Specifically the analysis is addressed to the derivation of criteria for the existence and uniqueness of the invariant probability measure and its regularity properties in the case of stochastic processes whose infinitesimal generator is uniformly elliptic or degenerate. The criteria are based on the definition of Lyapunov functions and the Hormander's rank bracket condition. Finally the criteria are employed for characterizing the invariant probability measure in some applications, including Kolmogorov-Fokker-Planck-type operators. |

Authors | C. Bianca, R. Sasportas |

Title | A one-dimensional mathematical model for the source reconstruction by the maximum entropy principle |

Journal | Applied Mathematics & Information Sciences 11 (2017), 1803-1809 |

Abstract
| This paper is concerned with the localization problem of a source belonging to a domain monitored by a network of detectors. A mathematical model is proposed within an inverse problem framework which is based on the maximum information entropy principle. Specifically the connection between the measurements released by the detectors and the sources is obtained by assuming that each detector has a visibility domain which is modeled by introducing a visibility function. A computational sensitivity analysis is performed on the number of detectors and on the visibility functions. The results are of great interest in the applied sciences. |

Authors | C. Bianca, A. Kombargi |

Title | On the modeling of the stock market evolution by means of the information-thermostatted kinetic theory |

Journal | Nonlinear Studies 24 (2017), 935-944 |

Abstract
| This paper is concerned with the modeling of the information which triggers the evolu- tion of the traders activity and the stock market. Specifically an inverse problem is defined within the framework of the continuous thermostatted kinetic theory for active particles coupled with the information theory. The inverse problem consists of a Volterra integral vector equation of the first kind. The well-posedness in the Hadamard sense is established by employing the maximum entropy principle. Research perspectives are also discussed within the paper. |

Authors | C. Bianca, G.M. Gallo, S. Motta |

Title | Tumor escape: A mathematical model |

Journal | IEEE BIBM (2017), 1401-1405 |

Abstract
| The immune system is a critical regulator of tumor biology with the capacity to support or inhibit tumor onset, growth, invasion and metastasis. Medical approaches designed to harness the immune system are the focus of several recent promising therapeutic approaches for cancer patients. However, not all tumors appear to respond to these immunomodulatory schemes. A major challenge for cancer immunotherapy, therefore, lies in understanding resistance mechanisms for selecting patients who are most likely to benefit. Moreover, tumors may also escape elimination either by recruiting immunosuppressive leukocytes which orchestrate a microenvironment that spoils the productivity of an anti-tumor immune response or underexpose cell-surface molecules. Therefore, predicting the clinical benefit of T cell immunotherapy is likely to require an understanding of each of these steps and their combinations as it relates to a patient’s individual tumor. Trial and errors approaches should be avoided in the clinical practice. Computer simulations based on mathematical model can help physician in evaluating in advance the effects of a given therapy on a class of patients. This paper describes a simple tumor escape mathematical model as first contribution toward a simulation based personalized medical treatment. |

Authors | C. Bianca, A. Kombargi |

Title | On the inverse problem for thermostatted kinetic models with application to the financial market |

Journal | Applied Mathematics & Information Sciences 11 (2017), 1463-1471 |

Abstract
| This paper is concerned with the coupling of the inverse problem theory with the thermostatted kinetic theory. Specifically an inverse problem is proposed where the data vector consists of m known measures, the data kernel is a mxn matrix which depends on the distribution function vector that is solution of the thermostatted kinetic theory model, and the unknown source or signal consists of a n-dimensional vector. In particular the paper focuses on the under-determined inverse problem, namely m < n, and the solution is obtained by employing the principle of maximum Shannon entropy of the information theory. Applications refer to the financial market and specifically to the derivation of the information which triggers the evolution of global stock market indexes. Future research directions are also discussed into the last section of the paper. |

Authors | C. Bianca |

Title | A new approach for the source problem based on the maximum entropy principle |

Journal | Nonlinear Studies 24 (2017), 715-723 |

Abstract
| This paper proposes a new mathematical approach that can be employed for the source localization problem. An inverse problem is thus established by conjecturing the relation between the sources and a measurement set data that is provided by a network of detectors deployed into a domain. Since the number of sources can be greater than the number of available measurements, the paper focuses on the under-determined inverse problem. Specifically the existence and uniqueness of the solution is investigated within the framework of the information theory and more precisely by employing the maximum information entropy principle. Applications and future research directions are outlined in the last section of the paper. |

Authors | C. Bianca |

Title | On the coupling of the thermostatted kinetic theory with the information theory |

Journal | Applied Mathematics & Information Sciences 11 (2017), 1767-1772 |

Abstract
| This paper deals with a further generalization of the continuous thermostatted kinetic theory for active particles. Specifically the interest focuses on the linking between the macroscopic data and the statistical evolution of the system. The connection between measurements and sources is established by defining an inverse problem based on the distribution vector function solution of the thermostatted kinetic framework. The inverse problem belongs to the class of ill-posed Volterra equations of the first kind considering that the number of sources can be greater of the number of measurements. The uniqueness of the solution is obtained by coupling the thermostatted kinetic theory with the information theory and more precisely with themaximum entropy principle of Jayne. Applications, which are discussed into the last section of the paper, refer to biological systems, vehicular traffic, crowds dynamics, and finance. |

Authors | M. Ben Amar, C. Bianca |

Title | Onset of nonlinearity in a stochastic model for auto-chemotactic advancing epithelia |

Journal | Nature Scientific Reports 6 (2016), 33849 |

Abstract
| We investigate the role of auto-chemotaxis in the growth and motility of an epithelium advancing on a solid substrate. In this process, cells create their own chemoattractant allowing communications among neighbors, thus leading to a signaling pathway. As known, chemotaxis provokes the onset of cellular density gradients and spatial inhomogeneities mostly at the front, a phenomenon able to predict some features revealed in in vitro experiments. A continuous model is proposed where the coupling between the cellular proliferation, the friction on the substrate and chemotaxis is investigated. According to our results, the friction and proliferation stabilize the front whereas auto-chemotaxis is a factor of destabilization. This antagonist role induces a fingering pattern with a selected wavenumber k_0. However, in the planar front case, the Galilean invariance of the experimental set-up gives also a mode at k = 0 and the coupling between these two modes in the nonlinear regime is responsible for the onset of a Hopf-bifurcation. The time-dependent oscillations of patterns observed experimentally can be predicted simply in this continuous non-linear approach. Finally the effects of noise are also investigated below the instability threshold. |

Authors | C. Bianca, A. Lemarchand |

Title | Miming the cancer-immune system competition by kinetic Monte Carlo simulations |

Journal | Journal of Chemical Physics 145 (2016), 154108 |

Abstract
| In order to mimic the interactions between cancer and the immune system at cell scale, we propose a minimal model of cell interactions that is similar to a chemical mechanism including autocatalytic steps. The cells are supposed to bear a quantity called activity that may increase during the interactions. The fluctuations of cell activity are controlled by a so-called thermostat. We develop a kinetic Monte Carlo algorithm to simulate the cell interactions and thermalization of cell activity. The model is able to reproduce the well-known behavior of tumors treated by immunotherapy: the first apparent elimination of the tumor by the immune system is followed by a long equilibrium period and the final escape of cancer from immunosurveillance. |

Authors | J. Liu, C. Bianca, L. Guerrini |

Title | Dynamical analysis of a computer virus model with delays |

Journal | Discrete Dynamics in Nature and Society 2016 (2016), 5649584 |

Abstract
| An SIQR computer virus model with two delays is investigated in the present paper. The linear stability conditions are obtained by using characteristic root method and the developed asymptotic analysis shows the onset of a Hopf bifurcation occurs when the delay parameter reaches a critical value. Moreover the direction of the Hopf bifurcation and stability of the bifurcating period solutions are investigated by using the normal form theory and the center manifold theorem. Finally, numerical investigations are carried out to show the feasibility of the theoretical results. |

Authors | M. Ben Amar, C. Bianca |

Title | Towards a unified approach in the modelling of fibrosis: A review with research perspectives |

Journal | Physics of Life Reviews 16 (2016), 61-85 |

Abstract
| Pathological fibrosis is the result of a failure in the wound healing process. The comprehension and the related modeling of the different mechanisms that trigger fibrosis are a challenge of many researchers that work in the field of medicine and biology. The modern scientific analysis of a phenomenon generally consists of three major approaches: theoretical, experimental, and computational. Different theoretical tools coming from mathematics and physics have been proposed for the modeling of the physiological and pathological fibrosis. However a complete framework is missing and the development of a general theory is required. This review aims at finding a unified approach in the modeling of fibrosis diseases that takes into account the different phenomena occurring at each level: molecular, cellular and tissue. Specifically by means of a critical analysis of the different models that have been proposed in the mathematical, computational and physical biology, from molecular to tissue scales, a multiscale approach is proposed, an approach that has been strongly recommended by top level biologists in the past decades. |

Authors | M. Ben Amar, C. Bianca |

Title | Multiscale modeling of fibrosis - What’s next? |

Journal | Physics of Life Reviews 16 (2016), 118-123 |

Abstract
| Following the commentary articles on the review [77], this reply aims at providing further issues on the possibility to develop a multiscale approach for the modeling of physiological and pathological fibrosis. Specifically this manuscript is devoted to the capability to employ new theoretical frameworks of the biomathematics, biophysics and bioinformatics for the derivation of a specific multiscale model including the possibility to validate the related model with experimental and empirical data. In particular comments on the mathematical control, frameworks with fractional-order, time delay, asymptotic methods, transport of drugs open the door to future investigations. |

Authors | C. Bianca, C. Dogbe |

Title | Recovering Navier-Stokes equations from asymptotic limits of the Boltzmann gas mixture equation |

Journal | Communications in Theoretical Physics 65 (2016), 553-562 |

Abstract
| This paper is devoted to the derivation of macroscopic fluid dynamics from the Boltzmann mesoscopic dynamics of a binary mixture of hard-sphere gas particles. Specifically the hydrodynamics limit is performed by employing different time and space scalings. The paper shows that, depending on the magnitude of the parameters which define the scaling, the macroscopic quantities (number density, mean velocity and local temperature) are solutions of the acoustic equation, the linear incompressible Euler equation and the incompressible Navier–Stokes equation. The derivation is formally tackled by the recent moment method proposed by [C. Bardos, et al., J. Stat. Phys. 63 (1991) 323] and the results generalize the analysis performed in [C. Bianca, et al., Commun. Nonlinear Sci. Numer. Simulat. 29 (2015) 240]. |

Authors | J. Riposo, C. Bianca |

Title | On the adjacency matrix of graphs: Principal eigenvector versus degree vector |

Journal | Nonlinear Studies 23 (2016), 365-378 |

Abstract
| This paper investigates on the relation between the principal eigenvector of the adjacency matrix and the degree vector of a graph. Specifically the analysis deals with the derivation of an upper bound, which only depends on the elements of the matrix, for the difference between the principal eigenvector and the degree vector. The results hold true for a general irreducible, nonnegative, symmetric matrix. |

Authors | C. Bianca, L. Guerrini |

Title | Loosing stability and exhibiting limit cycle as consequence of time delay introduction into a computer virus infection model |

Journal | Global and Stochastic Analysis 3 (2016), 1-10 |

Abstract
| This paper deals with analytical investigations on the possibility to model nonlinear dynamics emerging by the introduction of time delays into mathematical models based on differential equations. Specifically a delayed three-dimension ODE-based model is considered and an analytical sensitivity analysis on the time delays is performed. The results show that the magnitude of the time delay, considered as a parameter of the model, modifies the stability of the equilibrium points and induces the onset of a Poincaré- Andronov-Hopf bifurcation. Applications and future research directions are also discussed within the paper. |

Authors | BK Singh, C. Bianca |

Title | A new numerical approach for the solutions of partial differential equations in three-dimensional space |

Journal | Applied Mathematics & Information Sciences 10 (2016), 1663-1672 |

Abstract
| This paper deals with the numerical computation of the solutions of nonlinear partial differential equations in three-dimensional space subjected to boundary and initial conditions. Specifically, the modified cubic B-spline differential quadrature method is proposed where the cubic B-splines are employed as a set of basis functions in the differential quadrature method. The method transforms the three-dimensional nonlinear partial differential equation into a system of ordinary differential equations which is solved by considering an optimal five stage and fourth-order strong stability preserving Runge-Kutta scheme. The stability region of the numerical method is investigated and the accuracy and efficiency of the method are shown by means of three test problems: The three-dimensional space telegraph equation, the Van der Pol nonlinear wave equation and the dissipative wave equation. The results show that the numerical solution is in good agreement with the exact solution. Finally the comparison with the numerical solution obtained with some numerical methods proposed in the pertinent literature is performed. |

Authors | C. Bianca, C. Dogbe, A. Lemarchand |

Title | From cellular to tissue scales by asymptotic limits of thermostatted kinetic models |

Journal | The European Physical Journal Plus 131 (2016), 41 |

Abstract
| Tumor growth strictly depends on the interactions occurring at the cellular scale. In order to obtain the linking between the dynamics described at tissue and cellular scales, asymptotic methods have been employed, consisting in deriving tissue equations by suitable limits of mesoscopic models. In this paper, the evolution at the cellular scale is described by thermostatted kinetic theory that include conservative, nonconservative (proliferation, destruction and mutations), stochastic terms, and the role of external agents. The dynamics at the tissue scale (cell-density evolution) is obtained by performing a low- field scaling and considering the related convergence of the rescaled framework when the scaling parameter goes to zero. |

Authors | L. Guerrini, L. Gori, A. Matsumoto, M. Sodini, Z. Zhang, C. Bianca |

Title | Time delayed equations as models in nature and society |

Journal | Discrete Dynamics in Nature and Society 2016 (2016), 1245765 |

Abstract
| Editorial article |

Authors | XJ Yang, ST Mohyud-Din, C. Cattani, C. Bianca, A. Kilicman |

Title | Special issue on ''Fractional calculus and applications'' |

Journal | Journal of King Saud University - Science 28 (2016), 1-2 |

Abstract
| Editorial article |

Authors | C. Bianca, C. Dogbe |

Title | On the Boltzmann gas mixture equation: Linking the kinetic and fluid regimes |

Journal | Communications in Nonlinear Science and Numerical Simulation 29 (2015), 240-256 |

Abstract
| This paper aims at developing a new connection between the Boltzmann equation and the Navier-Stokes equation. Specifically the paper deals with the derivation of the macroscopic equations from asymptotic limits of the Boltzmann equation for a binary gas mixture of hard-sphere gases. By extending the methodology of the single-component gases case and by employing different time and space scalings, we show that it is possible to recover, under suitable technical assumptions, various fluid dynamics equations like the incompressible linearized and nonlinear Navier-Stokes equations, the incompressible linearized and nonlinear Euler equations. The novelty of this paper is the method that we propose, which differs from the Hilbert and Chapman-Enskog expansions. Future research directions are also discussed in the last section of the paper with special attention at the different scalings that can be employed in order to obtain equations presenting a ghost effect. |

Authors | C. Bianca, A. Lemarchand |

Title | Evaluation of reaction fluxes in stationary and oscillating far-from-equilibrium biological systems |

Journal | Physica A: Statistical Mechanics and its Applications 438 (2015), 1-16 |

Abstract
| The complex spatio-temporal structures that appear in chemical and biological systems require far-from-equilibrium conditions which may lead to the circulation of reaction fluxes. We investigate how time asymmetry of cross-correlation functions of concentration fluctuations may be exploited to determine reaction fluxes at the cellular level. Using simulations of the master equation as a reference, we show that, far from a bifurcation, the Langevin approach provides a reliable tool to compute analytical expressions for time correlation functions. Biochemical mechanisms associated with bistability and oscillations issued from a Hopf bifurcation or a saddle-node infinite period bifurcation are considered. We show that the blind use of the simple relation obtained when assuming a linear deterministic dynamics often leads to a poor estimation of the value of the reaction flux and even of its sign. |

Authors | C. Bianca, L. Guerrini, J. Riposo |

Title | A delayed mathematical model for the acute inflammatory response to infection |

Journal | Applied Mathematics & Information Sciences 9 (2015), 2775-2782 |

Abstract
| This paper deals with further developments on a mathematical model recently proposed for the modeling of the acute inflammatory response to infection or trauma. In particular in order to take into account that some interactions have not an immediate effect, we introduce time delays. Specifically the paper deals with the existence of steady states, determining the parameter regimes where the equilibrium points are stable, and the onset of Hopf bifurcation appears. Numerical simulations are performed with the main aim of supporting the analytical results and investigate further dynamics. |

Authors | J. Riposo, C. Bianca |

Title | On volatility variation in ARCH(1) and GARCH(1,1) continuous limits |

Journal | Nonlinear Studies 22 (2015), 359-371 |

Abstract
| The variance of many time series in biology and finance is not a constant function with respect to time. Different time-discrete statistical models have been proposed but recently timecontinuous limit models have been developed. This article is concerned with the analysis of volatility variation by employing the continuous limit of a time-discrete statistical models ARCH (AutoRegressive Conditionnal Heteroscedasticity) and GARCH (Generalized AutoRegressive Conditionnal Heteroscedasticity). Specifically, under some technical assumptions, we prove the uniqueness of the time-continuous processes ARCH(1)-M and GARCH(1,1)-M and we derive the related stationary probability distribution functions. Moreover by employing numerical simulations we show that the volatility variation is higher in the time-discrete GARCH(1,1) than ARCH(1). The results are of great interest in the financial markets. |

Authors | C. Bianca, J. Riposo |

Title | Mimic therapeutic actions against keloid by thermostatted kinetic theory methods |

Journal | The European Physical Journal Plus 130 (2015), 159 |

Abstract
| This paper deals with the modeling of a wound healing disease under a therapeutic action by employing the methods of the thermostatted kinetic theory for active particles. In particular, in order to test therapeutic actions against keloid formation and the possible development of a cancer, an external force field coupled to a Gaussian thermostat is introduced into a mathematical model recently proposed. Specifically the model depicts the competition of the immune system cells with a virus, the mutated fibroblast cells, and the cancer cells. Employing a computational analysis, the effects of three different external forces mimic therapeutic actions is analyzed: A vaccine for the virus, the PUVA therapy for the keloid and a vaccine for the cancer. The results are in agreement with the evidence that the sole action of the immune system is not sufficient to obtain a total depletion of keloid thus requiring the definition of a therapy. Further refinements and developments of the model are also discussed into the paper. |

Authors | C. Bianca, A. Lemarchand |

Title | Density evolution by the low-field limit of kinetic frameworks with thermostat and mutations |

Journal | Communications in Nonlinear Science and Numerical Simulation 20 (2015), 14-23 |

Abstract
| This paper is concerned with an asymptotic limit of a thermostatted kinetic framework which can be proposed for the modeling of complex biological and chemical systems where proliferative/destructive/mutative interactions and random velocity-jump processes occur. Specifically the macroscopic equations fulfilled by the local and global densities of the system are obtained by performing an asymptotic limit of a low-field rescaling of a kinetic framework with a thermostat and mutative interactions. |

Authors | C. Bianca, C. Dogbe |

Title | Mean-field limit of a microscopic individual-based model describing collective motions |

Journal | Journal of Nonlinear Mathematical Physics 22 (2015), 117-143 |

Abstract
| This paper is mainly concerned with a mean-field limit and long time behavior of stochastic microscopic interacting particles systems. Specifically we prove that a class of ODE modeling collective interactions in animals or pedestrians converges in the mean-field limit to the solution of a non-local kinetic PDE. The mathematical analysis, performed by weak measure solutions arguments, shows the existence of measure-valued solutions, asymptotic stability and chaos propagation that are relevant properties in the description of collective behaviors that emerge in animals and pedestrians motions. |

Authors | C. Bianca, C. Dogbe, A. Lemarchand |

Title | The role of nonconservative interactions in the asymptotic limit of thermostatted kinetic models |

Journal | Acta Applicandae Mathematicae 189 (2015), 1-24 |

Abstract
| This paper is concerned with the asymptotic analysis of space-velocity dependent thermostatted kinetic frameworks which include conservative, nonconservative and stochastic operators. The mathematical frameworks are integro-partial differential equations that can be proposed for the modeling of most phenomena occurring in biological and chemical systems. Specifically the paper focuses on the derivation of macroscopic equations obtained by performing a low-field and a high-field scaling into the thermostatted kinetic framework and considering the related convergence when the scaling parameter goes to zero. In the low-field limit, the macroscopic equations show diffusion with respect to both the space variable and a scalar variable that is introduced for the modeling of the strategy of the particle system. In the high-field limit, the macroscopic equations show hyperbolic behavior. The asymptotic analysis is also generalized to systems decomposed in various functional subsystems. |

Authors | C. Bianca, L. Guerrini |

Title | Hopf bifurcations in a delayed microscopic model of credit risk contagion |

Journal | Applied Mathematics & Information Sciences 9 (2015), 1493-1497 |

Abstract
| This paper is concerned with the proof of the existence of Hopf bifurcations in a mathematical model recently proposed in [T. Chen, X. Li, and J. He, Abstract and Applied Analysis 2014, 456764 (2014)] for understanding the complex stochastic dynamics phenomena of credit risk contagion in the financial market. Specifically the model consists in an ordinary differential equation with time-delay. Moreover, by using the normal form theory and center manifold argument, the stability, direction, and period of bifurcating periodic solutions are gained. |

Authors | C. Bianca |

Title | New research perspectives on thermostatted kinetic models |

Journal | Journal of Mathematics and Statistics 11 (2015), 16-20 |

Abstract
| This paper presents the new research directions that can be followed in the modeling of complex living systems by means of the tools of the thermostatted kinetic theory. The main aim is to invite the researchers that work in the field of applied mathematics to contribute to further developments of the theory with particular attention to applications in biology, crowds and swarm dynamics, economic and social systems |

Authors | C. Bianca, A. Lemarchand |

Title | Reaction-diffusion approach to somite formation |

Journal | IFAC-PapersOnLine (2015), 346-351 |

Abstract
| With the aim to propose minimal models of vertebrae formation, we present two kinds of reaction-diffusion models, the first one of clock-and-wavefront type and the second one of Turing type. A correspondence between the species of the reaction schemes and biologically relevant molecules known as morphogens will be proposed and the robustness of the spatial structures to internal fluctuations will be examined. The ability of the model of Turing type to reproduce experiments involving grafting of morphogen sources or sinks into embryos will be shown. |

Authors | C. Bianca, A. Lemarchand |

Title | Multiscale analysis of a retarded equation: From kinetic to macroscopic scale |

Journal | IFAC-PapersOnLine (2015), 656-660 |

Abstract
| This paper is devoted to the problem of linking the dynamics at kinetic scale, described by a retarded thermostatted kinetic equation, with the dynamics at macroscopic scale. Specically the macroscopic equation is derived by considering a scaling parameter into the underlying retarded equation and considering the related convergence when the scaling parameter goes to zero. The results show the onset of diffusion at macroscopic scale. |

Authors | C. Bianca, C. Dogbe, A. Lemarchand |

Title | Onset of hyperbolic macroscopic behavior in complex systems subjected to external agents |

Journal | Applied Mathematics & Information Sciences 9 (2015), 2477-2488 |

Abstract
| This paper deals with the derivation of hyperbolic equations from a space inhomogeneous thermostatted kinetic equation which can be proposed for the modeling of complex systems subjected to the external actions at the microscopic and macroscopic scales. The particles of the system are able to perform an activity which is modeled by introducing a specific variable. The derivation of the hyperbolic equations is obtained by performing different scalings into the time and space variables and letting the scaling parameter goes to zero. Applications and future research directions are discussed into the last section of the paper. |

Authors | C. Bianca, C. Dogbe |

Title | Kinetic models coupled with Gaussian thermostats: Macroscopic frameworks |

Journal | Nonlinearity 27 (2014), 2771-2803 |

Abstract
| This paper deals with the modelling of complex systems composed by a large number of elements grouped into different functional subsystems. The modelling framework is that of the thermostatted kinetic theory which consists in a set of nonlinear integro-differential equations. A source of nonlinearity is also the presence of the mathematical thermostat that ensures the control of the global energy of the system. Specifically this paper is devoted to the derivation of evolution equations for the macroscopic variables (density and momentum) from the underlying description at the microscopic scale delivered by the thermostatted kinetic models. To this aim, hyperbolic and parabolic type scalings of the thermostatted kinetic for active particles model are performed and the resulting macroscopic equations are obtained. Finally the asymptotic methods are applied to the relaxation model. |

Authors | C. Bianca, A. Lemarchand |

Title | Determination of reaction flux from concentration fluctuations near a Hopf bifurcation |

Journal | Journal of Chemical Physics 141 (2014), 144102 |

Abstract
| Small open chemical systems, typically associated with far-from-equilibrium, nonlinear stochastic dynamics, offer the appropriate framework to elucidate biological phenomena at the cellular scale. Stochastic differential equations of Langevin-type are employed to establish the relation between the departure from equilibrium and the time cross-correlation functions of concentration fluctuations for chemical species susceptible to oscillate. Except in the immediate vicinity of the Hopf bifurcation, the results are in agreement with simulations of the chemical master equation but always differ from the prediction obtained for linear deterministic dynamics. In general, the magnitude of the asymmetry of time correlation functions definitely depends on the reaction flux circulating in an open system but also on the details of the nonlinearities of deterministic dynamics. |

Authors | C. Bianca, A. Lemarchand |

Title | Temporal cross-correlation asymmetry and departure from equilibrium in a bistable chemical system |

Journal | Journal of Chemical Physics 140 (2014), 224105 |

Abstract
| This paper aims at determining sustained reaction fluxes in a nonlinear chemical system driven in a nonequilibrium steady state. The method relies on the computation of cross-correlation functions for the internal fluctuations of chemical species concentrations. By employing Langevin-type equations, we derive approximate analytical formulas for the cross-correlation functions associated with nonlinear dynamics. Kinetic Monte Carlo simulations of the chemical master equation are performed in order to check the validity of the Langevin equations for a bistable chemical system. The two approaches are found in excellent agreement, except for critical parameter values where the bifurcation between monostability and bistability occurs. From the theoretical point of view, the results imply that the behavior of cross-correlation functions cannot be exploited to measure sustained reaction fluxes in a specific nonlinear system without the prior knowledge of the associated chemical mechanism and the rate constants. |

Authors | C. Bianca, M. Ferrara, L. Guerrini |

Title | High-order moments conservation in thermostatted kinetic models |

Journal | Journal of Global Optimization 58 (2014), 389-404 |

Abstract | Recently the thermostatted kinetic framework has been proposed as mathematical model for studying nonequilibrium complex systems constrained to keep constant the total energy. The time evolution of the distribution function of the system is described by a nonlinear partial integro-differential equation with quadratic type nonlinearity coupled with the Gaussian isokinetic thermostat. This paper is concerned with further developments of this thermostatted framework. Specifically the term related to the Gaussian thermostat is adjusted in order to ensure the conservation of even high-order moments of the distribution function. The derived framework that constitutes a new paradigm for the derivation of specific models in the applied sciences, is analytically investigated. The global in time existence and uniqueness of the solution to the relative Cauchy problem is proved. Existence and moments conservation of stationary solutions are also performed. Suitable applications and research perspectives are outlined in the last section of the paper. |

Authors | C. Bianca, L. Guerrini, A. Lemarchand |

Title | Existence of solutions of a partial integro-differential equation with with thermostat and time delay |

Journal | Abstract and Applied Analysis 2014 (2014), 463409 |

Abstract
| This paper deals with the mathematical analysis of a retarded partial integro-differential equation that belongs to the class of thermostatted kinetic equations with time delay. Specifically the paper is devoted to the proof of the existence and uniqueness of mild solutions of the related Cauchy problem. The main result is gained by employing integration along the characteristic curves and successive approximations sequence arguments. Applications and perspective are also discussed within the paper. |

Authors | C. Bianca |

Title | How do mutative events modify moments evolution in thermostatted kinetic models? |

Journal | Communications in Nonlinear Science and Numerical Simulation 19 (2014), 2155-2159 |

Abstract
| This short communication aims at developing a thermostatted kinetic framework which includes conservative and nonconservative interactions. Specifically nonconservative interactions refer to proliferative/destructive and mutative events. The thermostatted kinetic framework is a set of autonomous partial integrodifferential equations with quadratic nonlinearity. How the moments evolution is modified by mutative interactions is explored in the present communication. Applications refer to the cancer-immune system competition. |

Authors | F. Castiglione, F. Pappalardo, C. Bianca, G. Russo, S. Motta |

Title | Modeling biology spanning different scales: An open challenge |

Journal | BioMed Research International 2014 (2014), 902545 |

Abstract
| It is coming nowadays more clear that in order to obtain a unified description of the different mechanisms governing the behavior and causality relations among the various parts of a living system, the development of comprehensive computational and mathematical models at different space and time scales are required. This is one of the most formidable challenges of modern biology characterized by the availability of huge amount of high throughput measurements. In this article we draw attention to the importance of multi-scale modeling in the framework of studies of biological systems in general and of the immune system in particular. |

Authors | C. Bianca and L. Guerrini |

Title | Existence of limit cycles in the Solow model with delayed-logistic population growth |

Journal | The Scientific World Journal 2014 (2014), 207806 |

Abstract
| This paper is devoted to the existence and stability analysis of limit cycles in a delayed mathematical model for the economy growth. Specifically the Solow model is further improved by inserting the time delay in the logistic population growth rate. Moreover by choosing the delay time as a bifurcation parameter, we prove that the system loses its stability and a Hopf bifurcation occurs when time delay passes through critical values. Finally numerical simulations are carried out for supporting the analytical results. |

Authors | C. Bianca, M. Ferrara, L. Guerrini |

Title | The asymptotic limit of an integro-differential equation modelling complex systems |

Journal | Izvestiya: Mathematics 78 (2014), 3-18 |

Abstract | This paper is devoted to the asymptotic analysis of a mathematical framework that has been recently proposed for the modeling of complex systems in the applied sciences under the action of an external force field. The framework consists in an integro-differential kinetic equation coupled with the Gaussian isokinetic thermostat. The asymptotic limit performed in the present paper by using the low-field scaling shows the emergence of diffusive behaviour at the macroscopic scale. |

Authors | C. Bianca, A. Lemarchand |

Title | A kinetic framework for modeling nonequilibrium biological and chemical systems |

Journal | Nonlinear Studies 21 (2014), 367-374 |

Abstract
| This paper deals with the introduction of nonconservative interactions in the thermostatted kinetic equations system, which is a set of autonomous partial integro-differential equations with quadratic nonlinearity. The time evolution of the solutions is obtained by considering three different operators: the operator which models conservative interactions, the operator that takes care of nonconservative interactions and the operator that allows the control of the moment evolution of the system. Applications refer to the modeling of complex biological and chemical systems. |

Authors | C. Bianca, C. Dogbe, L. Guerrini |

Title | A thermostatted kinetic framework with particle refuge for the modeling of tumor hiding |

Journal | Applied Mathematics & Information Sciences 8 (2014), 469-473 |

Abstract | Recently, thermostatted kinetic models has been proposed for the modeling of complex systems subjected to external force fields. This paper deals with the derivation of a generalized thermostatted kinetic framework which incorporates particles refuge. From the qualitative viewpoint, the global in time existence and uniqueness of the solution to the relative Cauchy problem, the time evolution of the moments of the solution, and the existence of stationary solutions are discussed. Applications refer to biological systems and especially to tumor-escape from immune system surveillance, indeed within this framework an abstract model for tumor-immune system competition is derived. Further research directions are outlined in the last section of the paper. |

Authors | C. Bianca, L. Guerrini, M. Ferrara, C. Udriste |

Title | Nonlinear Dynamics in Applied Sciences Systems: Advances and Perspectives |

Journal | Abstract and Applied Analysis 2014 (2014), 782657 |

Abstract | Editorial Article. |

Authors | C. Bianca |

Title | Existence of stationary solutions in kinetic models with Gaussian thermostats |

Journal | Mathematical Methods in the Applied Sciences 36 (2013), 1768–1775 |

Abstract
| The thermostatted kinetic framework has been recently proposed in [C. Bianca, Nonlinear Analysis: RealWorld Applications 13 (2012) 2593-2608] for the modeling of complex systems in the applied sciences under the action of an external force field that moves out of equilibrium the system. The framework consists in an integro-differential equation with quadratic nonlinearity coupled with the Gaussian isokinetic thermostat. This paper is concerned with the existence of stationary solutions proof. The main result is gained by fixed point and measure theory arguments. |

Authors | C. Bianca and L. Guerrini |

Title | On the Dalgaard-Strulik model with logistic population growth rate and delayed-carrying capacity |

Journal | Acta Applicandae Mathematicae 128 (2013), 39-48 |

Abstract | Recently Dalgaard and Strulik have proposed an energy model of capital accumulation based on the mathematical framework developed by Solow-Swan and coupled with Cobb-Douglas production function. The model is based on a constant rate of population growth assumption. The present paper, according to the analysis performed by Yukalov et al., improves the Dalgaard-Strulik model by introducing a logistic-type equation with delayed carrying capacity which alters the asymptotic stability of the relative steady state. Specifically, by choosing the time delay as a bifurcation parameter, it turns out that the steady state loses stability and a Hopf bifurcation occurs when time delay passes through critical values. The results are of great interest in the applied and theoretical economics. |

Authors | C. Bianca, M. Ferrara, L. Guerrini |

Title | The time delays effects on the qualitative behavior of an economic growth model |

Journal | Abstract and Applied Analysis 2013 (2013), 901014 |

Abstract
| A further generalization of an economic growth model is the main topic of this paper. Specifically the paper analyzes the effects on the asymptotic dynamics of the Solow model when two time delays are inserted: the time employed in order to the capital is used for production and the necessary time so that the capital is depreciated. The existence of a unique non-trivial positive steady state of the generalized model is proved and sufficient conditions for the asymptotic stability are established. Moreover the existence of a Hopf bifurcation is proved and, by using the normal form theory and center manifold argument, the explicit formulas which determine the stability, direction and period of bifurcating periodic solutions are obtained. Finally, numerical simulations are performed for supporting the analytical results. |

Authors | C. Bianca |

Title | Modeling complex systems with particles refuge by thermostatted kinetic theory methods |

Journal | Abstract and Applied Analysis 2013 (2013), 152174 |

Abstract
| This paper is concerned with the mathematical modeling of complex systems characterized by particles refuge. Specifically the paper focuses on the derivation and moments analysis of thermostatted kinetic frameworks with conservative and nonconservative interactions for closed and open complex systems at nonequilibrium. Applications and future research perspectives are discussed in the last section of the paper. |

Authors | C. Bianca |

Title | Controllability in Hybrid Kinetic Equations Modeling Nonequilibrium Multicellular Systems |

Journal | The ScientificWorld Journal 2013 (2013), 274719 |

Abstract
| This paper is concerned with the derivation of hybrid kinetic partial integrodifferential equations that can be proposed for the mathematical modeling of multicellular systems subjected to external force fields and characterized by nonconservative interactions. In order to prevent an uncontrolled time evolution of the moments of the solution, a control operator is introduced which is based on the Gaussian thermostat. Specifically, the analysis shows that the moments are solution of a Riccati-type differential equation. |

Authors | C. Bianca, M. Ferrara, L. Guerrini |

Title | Qualitative analysis of a retarded mathematical framework with applications to living systems |

Journal | Abstract and Applied Analysis 2013 (2013), 736058 |

Abstract
| This paper deals with the derivation and the mathematical analysis of an autonomous and nonlinear ordinary differential-based framework. Specifically the mathematical framework consists of a system of two ordinary differential equations: a logistic equation with a time lag and an equation for the carrying capacity here assumed time-dependent. The qualitative analysis refers to the stability analysis of the coexistence equilibrium and to the derivation of sufficient conditions for the existence of Hopf-bifurcations. The results are of great interest in living systems, including biological and economic systems. |

Authors | C. Bianca, F. Pappalardo, M. Pennisi, M.A. Ragusa |

Title | Persistence analysis in a Kolmogorov-type model for cancer-immune system competition |

Journal | AIP Conference Proceedings 1558 (2013), 1797-1800 |

Abstract | This paper is concerned with analytical investigations on the competition between cancer cells and immune system cells. Specifically the role of the B-cells and T-cells in the evolution of cancer cells is taken into account. The mathematical model is a Kolmogorov-type system of three evolution equations where the growth rate of the cells is described by logistic law and the response of B-cells and T-cells is modeled according to Holling type-II function. The stability analysis of equilibrium points is performed and the persistence of the model is proved. |

Authors | C. Bianca |

Title | Thermostatted kinetic models: Open problems |

Journal | J Appl Computat Math 2 (2013), doi:10.4172/2168-9679.1000e133 |

Abstract
| This editorial article is concerned with some open problems regarding the thermostatted kinetic theory for active particles. This framework refers to the modeling of complex systems composed by a large number of elements grouped into different functional subsystems. |

Authors | C. Bianca and C. Dogbe |

Title | A mathematical model for crowd dynamics: Multiscale analysis, fluctuations and random noise |

Journal | Nonlinear Studies 20 (2013), 281-305 |

Abstract | We propose and analyze a multiscale mathematical model that reproduces the predominant features of crowd dynamics by taking into account the distance among pedestrians. The Liouville equation, some ideas borrowed from kinetic theory and Grad limiting procedure are at the basis of the derivation of the model. Fluctuations and random noise (e.g. Brownian motion) are also considered. The asymptotic analysis shows that the probability distribution function of the crowd model, when converging, leads to standard kinetic models. Mono-kinetic descriptions are also investigated. Finally a momentum balance equation is readily stated and the macroscopic average velocity is obtained by averaging the mesoscopic description. The methodology used in this paper is based on particle, kinetic and hydrodynamic descriptions. |

Authors | C. Bianca, M. Ferrara and L. Guerrini |

Title | Hopf bifurcations in a delayed-energy-based model of capital accumulation |

Journal | Applied Mathematics & Information Sciences 7 (2013), 139-143 |

Abstract
| Building on a contribution by Dalgaard and Strulik [C.L. Dalgaard and H. Strulik, Resource and Energy Economics 33, 782 (2011)], this paper deals with the mathematical modelling for an economy viewed as a transport network for energy in which the law of motion of capital occurs with a time delay. By choosing time delay as a bifurcation parameter, it is proved that the system loses stability and a Hopf bifurcation occurs when time delay passes through critical values. An important scenario arising from the analysis is the existence of limit cycles generated by supercritical Hopf bifurcations. The results are of great interest for the analysis of the asymptotic economic growth. |

Authors | C. Bianca, M. Ferrara and L. Guerrini |

Title | The Cai model with time delay: Existence of periodic solutions and asymptotic analysis |

Journal | Applied Mathematics & Information Sciences 7 (2013), 21-27 |

Abstract | The economic growth model with endogenous labor shift under a dual economy proposed by Cai [Applied Mathematics Letters 21, 774-779 (2008)] is generalized in this paper by introducing a time delay in the physical capital. By choosing the delay as a bifurcation parameter, it is proved that the delayed model has unique nonzero equilibrium and a Hopf bifurcation is proven to exist as the delay crosses a critical value. Moreover the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated in this paper by applying the center manifold theorem and the normal form theory. |

Authors | C. Bianca |

Title | Thermostatted models - Multiscale analysis and tuning with real-world systems data |

Journal | Physics of Life Reviews 9 (2012), 418-425 |

Abstract | Following the commentary articles on the review [30], this reply aims at providing an efficient answer on the possibility to apply thermostatted kinetic equations for the modeling of immune system and related pathologies, vehicular traffic, crowds dynamics, economic systems, semiconductors devices. Specifically this manuscript is devoted to the possibility to perform a multiscale approach by using asymptotic methods of the kinetic theory existing in the pertinent literature. The possibility to tune the related equations with experimental and empirical data is the main core of this paper. |

Authors | C. Bianca |

Title | From physics to living systems - Applicable mathematical models |

Journal
| J Applied Computat Mathemat (2012), doi:10.4172/2168-9679.1000e123 |

Abstract | Editorial article |

Authors | C. Bianca |

Title | Thermostatted kinetic equations as models for complex systems in physics and life sciences |

Journal | Physics of Life Reviews 9 (2012), 359-399 |

Abstract | Statistical mechanics is a powerful method for understanding equilibrium thermodynamics. An equivalent theoretical framework for nonequilibrium systems has remained elusive. The thermodynamic forces driving the system away from equilibrium introduce energy that must be dissipated if nonequilibrium steady states are to be obtained. Historically, further terms were introduced, collectively called a thermostat, whose original application was to generate constant-temperature equilibrium ensembles. This review surveys kinetic models coupled with time-reversible deterministic thermostats for the modelling of large systems composed both by inert matter particles and living entities. The introduction of deterministic thermostats allows to model the onset of nonequilibrium stationary states that are typical of most real-world complex systems. The first part of the paper is focused on a general presentation of the main physical and mathematical definitions and tools: nonequilibrium phenomena, Gauss least constraint principle and Gaussian thermostats. The second part provides a review of a variety of thermostatted mathematical models in physics and life sciences, including Kac, Boltzmann, Jager-Segel and the thermostatted (continuous and discrete) kinetic for active particles models. Applications refer to semiconductor devices, nanosciences, biological phenomena, vehicular traffic, social and economics systems, crowds and swarms dynamics. |

Note
| Science Direct, TOP 25 Hottest Articles, Physics of Life Reviews, Ranking: 22/25, Year 2012 |

Authors | C. Bianca, F. Chiacchio, F. Pappalardo, M. Pennisi |

Title
| Mathematical modeling of the immune system recognition to mammary carcinoma antigen |

Journal
| BMC Bioinformatics 13 (2012), S21 |

Abstract
| The definition of artificial immunity, realized through vaccinations, is nowadays a practise widely developed in order to eliminate cancer disease. The present paper deals with an improved version of a mathematical model recently analyzed and related to the competition between immune system cells and mammary carcinoma cells under the action of a vaccine (Triplex). The model describes in detail both the humoral and cellular response of the immune system to the tumor associate antigen and the recognition process between B cells, T cells and antigen presenting cells. The control of the tumor cells growth occurs through the definition of different vaccine protocols. The performed numerical simulations of the model are in agreement with in vivo experiments on transgenic mice. |

Authors | C. Bianca |

Title | Onset of nonlinearity in thermostatted active particles models for complex systems |

Journal | Nonlinear Analysis: Real World Applications 13 (2012), 2593-2608 |

Abstract
| This paper is concerned with the derivation of a new discrete general framework of the kinetic theory, suitable for the modeling of complex systems under the action of an external force field and constrained to kept constant the mass or density, and the kinetic or activation energy. The resulting model relies on the interactions of single individuals within the population and is expressed by means of nonlinear ordinary or partial integrodifferential equations. The global in time existence and uniqueness of the solution to the relative Cauchy problem are proved for which the density and the energy of the solution are preserved. A critical analysis, proposed in the last part of the paper, outlines suitable applications and research perspectives. /td> |

Note
| Science Direct, TOP 25 Hottest Articles, Nonlinear Analysis: Real World Applications, Ranking: 2/25, Year 2012 |

Authors | C. Bianca |

Title | Mathematical modeling of crowds dynamics: Complexity and kinetic approach |

Journal
| Nonlinear Studies 19 (2012), 345-354 |

Abstract
| This paper deals with the kinetic theory framework for the modeling of the complex dynamics of crowds constituted by a large number of individuals (pedestrians) that interact in a nonlinear fashion in a domain with and without obstacles. The kinetic frameworks are coupled with the stochastic games theory and Gaussian thermostats in order to model the interactions which occur at microscopic scale. After introducing the mathematical structures for continuous and granular flows, the paper outlines suitable applications and research perspectives. |

Authors | C. Bianca and M. Pennisi |

Title | The triplex vaccine effects in mammary carcinoma: A nonlinear model in tune with simtriplex |

Journal | Nonlinear Analysis: Real World Applications 13 (2012), 1913-1940 |

Abstract
| This paper deals with the mathematical modeling of the mammary carcinoma-immune system competition elicited by an external stimulus represented by three different protocols of the triplex vaccine [De Giovanni, et. al., Cancer Research, 64 (2004)]. The presented model is composed of nonlinear ordinary differential equations based on parameters and cell populations. A qualitative analysis of the asymptotic behavior of the model and numerical simulations are able to depict preclinical experiments on transgenic mice in tune with the simtriplex model [Pappalardo, et. al., Bioinformatics, 21 (2005)]. The results are of great interest both in the applied and theoretical sciences. |

Note
| Science Direct, TOP 25 Hottest Articles, Nonlinear Analysis: Real World Applications, Ranking: 1/25, Year 2012 |

Authors | C. Bianca |

Title | Modeling complex systems by functional subsystems representation and thermostatted-KTAP methods |

Journal | Applied Mathematics & Information Sciences 6 (2012), 495-499 |

Abstract
| In the last two decades the mathematical modeling of real-world complex systems has received much attention. These systems are composed by a high number of interacting entities which are able to express a specific strategy. In order to reduce complexity, the whole system is decomposed in different subsystems that are sets of heterogeneous entities having the ability of expressing the same function. The microscopic interactions among the functional subsystems generate the emerging behaviors that are typical of the complex systems. This paper is concerned with suitable developments of the methods of mathematical kinetic theory for active particles for the modeling of complex systems splitted in functional subsystems and constrained to maintain constant some macroscopic quantities. |

Authors | C. Bianca |

Title | Thermostatted kinetic models for complex systems under microscopic external fields |

Journal | Mathematics in Engineering, Science and Aerospace 3 (2012), 225-238 |

Abstract
| Complex systems in the life and applied sciences evolve in time as consequence of the microscopic interactions occurring among the elements constituting the system. In real-world systems, the evolution of the system is usually due to both external actions at the macroscopic scale and interactions with the outer environment at the microscopic scale. This paper is concerned with the development and analysis of mathematical frameworks of the thermostatted kinetic theory for active particles for complex systems subjected both by macroscopic and microscopic external fields (open systems). These new classes of equations constitute a background paradigm for the derivation of specific models for self-organized biological aggregations, traffic-like movement, collective behaviors observed in animal and human communities and economical sciences. |

Authors | C. Bianca |

Title | Complex dynamic systems: Nonlinear methods, mathematical models and thermodynamics |

Journal | Mathematics in Engineering, Science and Aerospace 3 (2012), 221-224 |

Abstract
| This note introduces the topic of the papers published in this special issue devoted to the modeling and analysis of complex systems in the applied sciences. The study of complex systems in a unified framework has become recognized in recent years as a new scientific discipline, the ultimate of interdisciplinary fields. It is strongly rooted in the advances that have been made in diverse fields where complex dynamic arises, among others: physics, chemistry, biology, sociology, psychology, economics, anthropology, philosophy. |

Authors | C. Bianca |

Title | An existence and uniqueness theorem for the Cauchy problem for thermostatted-KTAP models |

Journal | Int. Journal of Math. Analysis 6 (2012), 813-824 |

Abstract | This paper is concerned with an integro-differential equation referred to as the thermostatted kinetic theory for active particles framework, which models complex systems under the action of an external force field, and constrained to kept constant the total energy (activity). The global in time existence and uniqueness of the solution to the relative Cauchy problem are proved for which the density and the energy of the particles are preserved. |

Authors | C. Bianca |

Title | Kinetic theory for active particles modelling coupled to Gaussian thermostats |

Journal | Applied Mathematical Sciences 6 (2012), 651-660 |

Abstract | This paper deals with the kinetic theory for active particles modelling of complex systems, under the action of a velocity-dependent force field, and constrained to kept constant the total kinetic energy. A further development of the mathematical theory is here proposed, which consists in the external force-Gaussian isokinetic thermostat coupling with the intent to preserve the kinetic energy of the system during the motion. The relative framework constitutes the paradigm for the derivation of specific models in the applied mathematical sciences, e.g., the chemotaxis phenomenon in biological systems. |

Authors | C. Bianca |

Title | On the existence of periodic orbits in nonequilibrium Ehrenfest gas |

Journal
| Int. Math. Forum 7 (2012), 221-232 |

Abstract
| Recently in a nonequilibrium polygonal billiard modelling simple microporous membranes, know as the modified Ehrenfest gas, has been established that the point-particle dynamics displays both chaotic (positive Lyapunov exponent) and nonchaotic steady states with a quite peculiar sensitive dependence on the electric field and on the geometry parameters. The existence of chaotic behavior allows to estimate chaotic averages by means of periodic orbit theory provided that periodic orbits exist. This paper investigates on the existence of periodic orbits as function of the geometry parameters. The results have both applied (e.g., nanotechnological, biological) and theoretical interest. |

Authors | C. Bianca and M. Pennisi |

Title | Immune system modelling by top-down and bottom-up approaches |

Journal | Int. Math. Forum 7 (2012), 109-128 |

Abstract | The biological immune system is a complex adaptive system that constitutes the defence mechanism of higher level organisms to micro organismic threats. There are lots of benefits for building an artificial (mathematical, physical or computational) model of the immune system. Medical researchers can use immune system simulation in drug research or to test hypotheses about the infection process. Given the wide range of uses for immune simulation and the difficulty of the task, it is useful to know what research has been conducted in this area. This paper provides a survey of the literatures in this field comparing and analyzing some of the existing approaches and models. |

Authors | C. Bianca |

Title | Mathematical modelling for keloid formation triggered by virus: Malignant effects and immune system competition |

Journal | Math. Models Methods Appl. Sci. 21 (2011), 389-419 |

Abstract | This paper deals with the modelling of genetic mutations, which occur in almost all cells of a living system. The mutated cells display different stages of cancer progression and are contrasted by the action of the immune system cells. This investigation can be of interest in the evolutionary dynamics of cellular systems since the selective pressure on the mutated cells exerted by the immune system is analyzed. The proposed mathematical model is developed by means of the tools of the kinetic theory of active particles. Numerical simulations, obtained considering different values of the parameters in the model, show different emerging behaviors that are typical of the cancer-immune system competition. |

Authors | C. Bianca and L. Fermo |

Title | Bifurcation diagrams for the moments of a kinetic type model of keloid-immune system competition |

Journal | Computers & Mathematics with Applications 61 (2011), 277-288 |

Abstract
| The mathematical modelling of the keloid disease triggered by a virus has been recently investigated by one of the authors, Bianca (2011) [18], where it was shown that the model is able to depict the emerging behaviours which occur during the keloid formation. This paper deals with further numerical investigations of that model related to the bifurcation analysis of the measurable macroscopic variables associated to each functional subsystem. It is shown that there exists a critical value of a bifurcation parameter separating situations where the immune system controls the keloid formation from those where malignant effects are not contrasted. |

Authors | C. Bianca, M. Pennisi, S. Motta, and M.A. Ragusa |

Title
| Immune system network and cancer vaccine |

Journal
| AIP Conf. Proc. 1389 (2011), 945-948 |

Abstract
| This paper deals with the mathematical modelling of the immune system response to cancer disease, and specifically with the treatment of the mammary carcinoma in presence of an immunoprevenction vaccine. The innate action of the immune system network, the external stimulus represented by repeated vaccine administrations and the competition with cancer are described by an ordinary differential equations-based model. The mathematical model is able to depict preclinical experiments on transgenic mice. The results are of great interest both in the applied and theoretical sciences. |

Authors | C. Bianca and M. Delitala |

Title | On the modelling of genetic mutations and immune system competition |

Journal | Computers & Mathematics with Applications 61 (2011), 2362-2375 |

Abstract
| This paper deals with the modelling at the cellular scale of a wound healing disease, the keloid, which may provoke onset of malignant cells with higher progression feature, thus generating cells with heterogeneous phenotype. According to medical hypothesis is assumed that viruses and the genetic susceptibility of patients are the main causes that trigger the formation. The mathematical model is developed by means of the tools of the kinetic theory for active particles. The competition of the immune system cells with viruses, keloid ¯broblast cells, and malignant cells is taken into account. Numerical simulations, obtained considering the sensitivity analysis of the parameters in the model, show the emerging phenomena that are typical of this disease. |

Authors | C. Bianca and V. Coscia |

Title | On the coupling of steady and adaptive velocity grids in vehicular traffic modelling |

Journal | Applied Mathematics Letters 24 (2011), 149-155 |

Abstract | This paper deals with the derivation and the analysis of a new mathematical model for vehicular traffic along a one-way road obtained by coupling of an uniform and an adaptive discretization of the velocity variable in the framework of the kinetic theory. Interactions are modelled by stochastic games where the output of interactions depends on the local density and is not linearly additive. |

Authors | N. Bellomo, C. Bianca, and V. Coscia |

Title | On the modeling of crowd dynamics: An overview |

Journal
| Bol. Soc. Esp. Mat. Apl. (Sema Journal) 54 (2011), 25-46 |

Abstract | This paper presents a critical analysis of the mathematical literature concerning the modeling of the complex living system of the crowds. The presentation is focused on the representation scales (microscopic, kinetic, and macroscopic) and on the mathematical frameworks that can be used for the modeling approach. The existing literature is critically analyzed and focused on research perspectives in view of a unified modeling strategy. |

Authors | C. Bianca |

Title | Weyl-flow and conformally symplectic structure of thermostatted billiards: The problem of the hyperbolicity |

Journal | Nonlinear Analysis: Hybrid Systems 5 (2011), 32-51 |

Abstract | The highly complex nature of the transport in thermostatted billiards has been of interest in the last few decades because of industrial and medical applications. The onset of hyperbolic dynamics (deterministic chaos) in such a billiard has evidenced an interesting stabilization of the transport properties, especially in microporous media. Recently, different mathematical methods have been developed for establishing hyperbolicity in thermostatted billiards, among these, the Weyl-flow and the conformally symplectic structure techniques. This paper deals with analytical investigations on the possible hyperbolic nature of two thermostatted billiards: The nonequilibrium Ehrenfest gas (NEEG) and the pump model (PM). Despite numerical investigations support the idea of their dissipative dynamics, the hyperbolicity of these billiards has not been yet established. The analysis developed in this paper shows how the Weyl-flow technique is failed for NEEG, revealing the necessity to develop new strategies in order to obtain hyperbolicity. On the contrary, we prove that the PM has a conformally symplectic structure, which is the basis for establishing the hyperbolicity of such a hybrid dynamical system. |

Authors | N. Bellomo, C. Bianca, and M.S. Mongiovi; |

Title
| On the modeling of nonlinear interactions in large complex systems |

Journal | Applied Mathematics Letters 23 (2010), 1372-1377 |

Abstract
| This work deals with the modeling of large systems of interacting entities in the framework of the mathematical kinetic theory for active particles. The contents are specifically focused on the modeling of nonlinear interactions which is one of the most important issues in the mathematical approach to modeling and simulating complex systems, and which includes a learning-hiding dynamics. Applications are focused on the modeling of complex biological systems and on immune competition. |

Authors | C. Bianca, |

Title | On the mathematical transport theory in microporous media: The billiard approach |

Journal
| Nonlinear Analysis: Hybrid Systems 4 (2010), 699-735 |

Abstract | This paper is an expository of the main dynamical properties of billiards, which depend on the shape of the walls of the container, and the recent developments like the introduction of an external field, which mimic the coupling with a thermostat. The class of dynamical system dealt with in this paper exhibits characteristics of hybrid systems as it links discrete and continuous, deterministic and stochastic dynamics. The contents are focused on applications. Specifically, transport dynamics in highly-confined regions has been of interest in the last few decades because of industrial and medical applications. Aspects of confined transport remain elusive, considering that in microporous membranes, whose size pores is about that of the molecules, the transport is sometimes ballistic, and sometimes diffusive. The classical kinetic and macroscopic approach can not be directly applied because collisions of particle fluid with walls prevail. The microscopic mathematical billiard theory can be applied as a mathematical tool since the interstices between obstacles can be considered as the pores of the membranes. |

Authors | A. Bellouquid and C. Bianca, |

Title | Modelling aggregation-fragmentation phenomena from kinetic to macroscopic scales |

Journal | Math. Comput. Modelling 52 (2010), 802-813 |

Abstract | This paper deals with the modelling of aggregation and/or fragmentation physical phenomena for large systems of interacting living entities in the framework of the mathematical kinetic theory for active particles. After introducing various mathematical structures in terms of systems of nonlinear integro-differential equations with quadratic type nonlinearities and variable number of equations, the relative qualitative analysis of the initial value problem is presented. Finally the paper deals with the derivation of macroscopic equations based on the underlying description at the microscopic scale delivered by the kinetic theory models. |

Authors | C. Bianca, |

Title | Thermostatted planar billiards as aimple models of mass transport in microporous media |

Journal | Communications in Applied and Industrial Mathematics 1 (2010), 22-40 |

Abstract
| The development and analysis of models of transport of matter is necessary to describe the anomalous transport phenomena in various applications, including some of nanotechnological interest, like the manufacturing of microporous membranes whose size pores is about that of the molecules. Nevertheless, chaotic dynamics has evidenced an interesting stabilization of the transport properties in microporous media. The small pores do not allow to apply directly neither the fluid dynamics laws nor the kinetic theories. In this context thermostatted planar billiards can be proposed as models since the interstices between obstacles can be considered as the pores of the membranes. We will show some results of both applied and theoretical interest. |

Authors | C. Bianca, |

Title | On the modelling of space dynamics in the kinetic theory for active particles |

Journal | Math. Comput. Modelling 51 (2010), 72-83 |

Abstract | This paper proposes an analysis of the modelling of space dynamics focused on a general class of models of the kinetic theory for active particles in space homogeneity. Various deterministic and stochastic developments are treated and referred to specific applications. These new classes of equations present different aspects of hybrid characteristics coupling deterministic and stochastic structures, as well as continuous and discrete variables, and constitute a background paradigm for the derivation of models whose qualitative properties are analyzed referring to modelling of complex systems in life and applied sciences. |

Authors | C. Bianca, |

Title | Modelli di biliardi caotici e poligonali per lo studio del trasporto in mezzi microporosi |

Journal
| La Matematica nella Societa' e nella Cultura, Rivista dell'Unione Matematica Italiana, Serie I, Vol. II, Agosto 2009, 203-206 |

ISBN
| 1972-7356 |

Authors | N. Bellomo, C. Bianca, and M. Delitala, |

Title | Complexity analysis and mathematical tools towards the modelling of living systems |

Journal | Physics of Life Reviews 6 (2009), 144-175 |

Abstract | This paper is a review and critical analysis of the mathematical kinetic theory of active particles applied to the modelling of large living systems made up of interacting entities. The first part of the paper is focused on a general presentation of the mathematical tools of the kinetic theory of active particles. The second part provides a review of a variety of mathematical models in life sciences, namely complex social systems, opinion formation, evolution of epidemics with virus mutations, and vehicular traffic, crowds and swarms. All the applications are technically related to the mathematical structures reviewed in the first part of the paper. The overall contents are based on the concept that living systems, unlike the inert matter, have the ability to develop behaviour geared towards their survival, or simply to improve the quality of their life. In some cases, the behaviour evolves in time and generates destructive and/or proliferative events. |

Authors | C. Bianca and S. Motta, |

Title | The MWF method: A convergence theorem for homogenous one-dimensional case |

Journal
| Computers & Mathematics with Applications 58 (2009), 579-588 |

Abstract | The MWF numerical method for kinetic equations was presented by S. Motta and J. Wick in 1992 and recently extended by the authors to systems of kinetic equations. The basic idea of the method consists in rewriting the kinetic equation in a conservation law in divergence form, redefining the collisions as a flux and formally to transform the problem into a collisionless one. In all tested cases, the numerical results are in agreement with the exact solutions but a convergence proof of the method, to the best of our knowledge, is missing. In this paper we present our investigation on the sufficient conditions that the collision operator may satisfy, to guarantee a convergence proof of the method in the homogeneous one-dimensional case. This investigation is of both theoretical and applied interest. |

Authors | C. Bianca, F. Pappalardo, and S. Motta, |

Title | The MWF method for kinetic equations system |

Journal
| Computers & Mathematics with Applications 57 (2009), 831-840 |

Abstract | Many physical or biological phenomena deal with the dynamics of interacting entities. These class of phenomena are well described in physics, using a kinetic approach based on Boltzmann equation. A Generalized Kinetic theory has been proposed to extend this approach to biological scenarios. An analytical solution of Boltzmann equation can be found only in very simple cases, so numerical methods become extremely relevant. The particle method is a class of numerical methods used to find a numerical solution of Boltzmann equations. The MWF-method for kinetic equations was firstly proposed by S. Motta and J. Wick in 1992. Here, we show that the MWF-method can be extended to system of Boltzamm equations. |

Authors | C. Bianca and L. Rondoni, |

Title | The nonequilibrium Ehrenfest gas: A chaotic model with flat obstacles? |

Journal
| Chaos 19, 013121 (2009) |

Abstract
| It is known that the nonequilibrium version of the Lorentz gas a billiard with dispersing obstacles [Ya. G. Sinai, Russ. Math. Surv. 25, 137 1970], electric field, and Gaussian thermostat is hyperbolic if the field is small [N. I. Chernov, Ann. Henri Poincare 2, 197 2001]. Differently the hyperbolicity of the nonequilibrium Ehrenfest gas constitutes an open problem since its obstacles are rhombi and the techniques so far developed rely on the dispersing nature of the obstacles [M. P. Wojtkowski, J. Math. Pures Appl. 79, 953 2000]. We have developed analytical and numerical investigations that support the idea that this model of transport of matter has both chaotic positive Lyapunov exponent and nonchaotic steady states with a quite peculiar sensitive dependence on the field and on the geometry, not observed before. The associated transport behavior is correspondingly highly irregular, with features whose understanding is of both theoretical and technological interests. © 2009 American Institute of Physics. |

Authors | O. Jepps, C. Bianca, and L. Rondoni, |

Title | Onset of diffusive behaviour in confined transport systems |

Journal
| Chaos 18, 013127 (2008) |

Abstract
| We investigate the onset of diffusive behavior in polygonal channels for disks of finite size, modeling simple microporous membranes. It is well established that the point-particle case displays anomalous transport, because of slow correlation decay in the absence of defocusing collisions. We investigate which features of point-particle transport survive in the case of finite-sized particles which undergo defocusing collisions. A similar question was investigated by Lansel, Porter, and Bunimovich [Chaos 16, 013129 2006], who found that certain integrals of motion and multiple ergodic components, characteristic of the point-particle case, remain in ''mushroom''-like systems with few finite-sized particles. We quantify the time scales over which the transport of disks shows features typical of the point particles, or is driven toward diffusive behavior. In particular, we find that interparticle collisions drive the system toward diffusive behavior more strongly than defocusing boundary collisions. We illustrate how, and at what stage, typical thermodynamic behavior consistent with kinetic theory is observed, as particle numbers grow and mean free paths diminish. These results have both applied e.g., nanotechnological and theoretical interest. |

Note
| It has been selected for the April 7, 2008 issue of ''Virtual Journal of Nanoscale Science & Technology'', edited by the American Institute of Physics and the American Physical Society |

Scientific Books

Authors | J. Riposo and C. Bianca |

Title | Mathematical and Computational Methods in Biology and Finance |

Editor
| LAP LAMBERT Academic Publishing, Germany, (2015) |

ISBN-13
| 978-3-659-79624-1 |

ISBN-10 | 365-9796-24-7 |

Authors | C. Bianca |

Title | Mathematical Billiards and Applications |

Editor
| LAP LAMBERT Academic Publishing, Germany, (2012) |

ISBN-13
| 978-3-8484-4914-9 |

ISBN-10 | 981-4340-53-7 |

Authors | C. Bianca and N. Bellomo |

Title | Towards a Mathematical Theory of Multiscale Complex Biological Systems |

Editor | World Scientific, Series in Mathematical Biology and Medicine, (2011) |

ISBN-13 | 978-981-4340-53-3 |

ISBN-10
| 981-4340-53-7 |

Teaching Books

Authors | C. Bianca |

Title | Algèbre I, Grandes Ecoles |

Editor
| Clut, Torino (2020) |

ISBN | 9788879924603 |

Authors | C. Bianca |

Title | Analyse I, Grandes Ecoles, Partie II |

Editor
| Clut, Torino (2019) |

ISBN | 9788879924559 |

Authors | C. Bianca |

Title | Analyse I, Grandes Ecoles, Partie I |

Editor
| Clut, Torino (2019) |

ISBN | 9788879924467 |

Authors | C. Bianca, L. Mazzi, |

Title | Pillole di Analisi Matematica II |

Editor
| Clut, Torino (2014) |

ISBN | 88-799-2362-0 |

Authors | C. Bianca |

Title | Analisi Matematica II in Test Svolti e Proposti |

Editor
| Clut, Torino (2012) |

ISBN | 88-799-2317-0 |

Authors | C. Bianca, F. Perri, |

Title | Chi ha Paura dell'Analisi...delle Successioni e Serie? |

Editor
| Pitagora Editrice, Bologna (2011) |

ISBN | 88-371-1857-0 |

Authors | C. Bianca and F. Perri, |

Title | Chi ha Paura dell'Analisi... Matematica I? |

Editor
| Pitagora Editrice, Bologna (2008) |

ISBN | 88-371-1753-1 |

Scientific Articles in Peer-Reviewed Conference Proceedings

Authors | A. Lemarchand, C. Bianca, |

Title
| Time asymmetry of cross-correlation functions as a signature of non equilibrium steady states (Chapter of Book) |

Published in | Proceedings of “The International Symposium on Mathematical and Computational Biology, BIOMAT 2014” |

World Scientific (2015), pp. 26-45, ISBN 978-981-4667-93-7 | |

Authors | M. Pennisi, C. Bianca, F. Pappalardo, and S. Motta, |

Title
| Compartmental mathematical modeling of immune system-melanoma competition |

Published in | Proceedings of ''The 11(th) International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2011'' |

(2011), ISBN 978-84-614-6167-7, pp. 1678-1682 | |

Authors | C. Bianca, M. Pennisi, and S. Motta, |

Title | The MWF method for kinetic models: An overview and research perspective |

Published
in | Proceedings of ''The 11(th) International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2011'' |

(2011), ISBN 978-84-614-6167-7, pp. 930-934 | |

Authors | M. Pennisi, C. Bianca, F. Pappalardo, and S. Motta, |

Title
| Modeling artificial immunity against mammary carcinoma |

Published in | Proceedings of ''The 10(th) International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2010'' |

(2010), ISBN 978-84-613-5510-5, pp. 753-756 | |

Authors | C. Bianca and L. Rondoni, |

Poster Title | Thermostatted planar billiards as simple model of mass transport in microporous membranes |

Published in | Proceedings of ''The 10(th) Experimental Chaos Conference, ECC10'' |

(2008), ISBN 978-88-7751-282-6 | |

Newspaper Articles

Authors | C. Bianca, |

Title | Dal Kosovo al Vaccarini |

Journal | La Sicilia , April 20th (1999) |