Qualitative analysis of kinetic-based models for tumor-immune system interaction

A mathematical model, based on a mesoscopic approach, describing the competition between tumor cells and immune system in terms of kinetic integro-differential equations is presented. Four interacting populations are considered, representing, respectively, tumors cells, cells of the host environment, cells of the immune system, and interleukins, which are capable to modify the tumor-immune system interaction and to contribute to destroy tumor cells. The internal state variable (activity) measures the capability of a cell of prevailing in a binary interaction. Under suitable assumptions, a closed set of autonomous ordinary differential equations is then derived by a moment procedure and two three-dimensional reduced systems are obtained in some partial quasi-steady state approximations. Their qualitative analysis is finally performed, with particular attention to equilibria and their stability, bifurcations, and their meaning. Results are obtained on asymptotically autonomous dynamical systems, and also on the occurrence of a particular backward bifurcation

A dual approach for positive T–S fuzzy controller design and its application to cancer treatment under immunotherapy and chemotherapy

This study proposes an effective positive control design strategy for cancer treatment by resorting to the combination of immunotherapy and chemotherapy. The treatment objective is to transfer the initial number of tumor cells and immune–competent cells from the malignant region into the region of benign growth where the immune system can inhibit tumor growth. In order to achieve this goal, a new modeling strategy is used that is based on Takagi–Sugeno. A Takagi-Sugeno fuzzy model is derived based on the Stepanova nonlinear model that enables a systematic design of the controller. Then, a positive Parallel Distributed Compensation controller is proposed based on a linear co-positive Lyapunov Function so that the tumor volume and administration of the chemotherapeutic and immunotherapeutic drugs is reduced, while the density of the immune-competent cells is reached to an acceptable level. Thanks to the proposed strategy, the entire control design is formulated as a Linear Programming problem. Finally, the simulation results show the effectiveness of the proposed control approach for the cancer treatment

Analysis and design of quadratic parameter varying (QPV) control systems with polytopic attractive region

© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper proposes a gain-scheduling approach for systems with a quadratic structure. Both the stability analysis and the state-feedback controller design problems are considered for quadratic parameter varying (QPV) systems. The developed approach assesses/enforces the belonging of a polytopic region of the state space to the region of attraction of the origin, and relies on a linear matrix inequality (LMI) feasibility problem. The main characteristics of the proposed approach are illustrated by means of examples, which confirm the validity of the theoretical results.Peer ReviewedPostprint (author's final draft

Mesenchymal Stromal Cells: Emerging Roles in Bone Metastasis:Emerging Roles in Bone Metastasis

Bone metastasis is the most advanced stage of many cancers and indicates a poor prognosis for patients due to resistance to anti-tumor therapies. The establishment of metastasis within the bone is a multistep process. To ensure survival within the bone marrow, tumor cells must initially colonize a niche in which they can enter dormancy. Subsequently, reactivation permits the proliferation and growth of the tumor cells, giving rise to a macro-metastasis displayed clinically as a bone metastatic lesion. Here, we review the evidences that suggest mesenchymal stromal cells play an important role in each of these steps throughout the development of bone metastasis. Similarities between the molecular mechanisms implicated in these processes and those involved in the homeostasis of the bone indicate that the metastatic cells may exploit the homeostatic processes to their own advantage. Identifying the molecular interactions between the mesenchymal stromal cells and tumor cells that promote tumor development may offer insight into potential therapeutic targets that could be utilized to treat bone metastasis

Fractional derivative models for the spread of diseases

This thesis considers the mathematical modelling of disease, using fractional differential equations in order to provide a tool for the description of memory effects. In Chapter 3 we illustrate a commensurate fractional order tumor model, and we find a critical value of the fractional derivative dependent on the parameter values of the model. For fractional derivatives of orders less than the critical value an unstable equilibrium point of the system becomes stable. In order to show changes in the observed areas of attraction of two stable points in the system, we then consider a fractional order SIR epidemic model and investigate the change from a monostable to a bistable system.;Chapter 4 considers a model for virus dynamics where the fractional orders for populations are different, called an incommensurate system. An approximate analytical solution for the characteristic equation of the incommensurate model is found when the different fractional orders are similar and close to the critical value of the fractional order of the commensurate system. In addition, the instability boundary is found as a function of both parameters. A comparison between analytical and numerical results shows the high accuracy of this approximation.;Chapter 5 consists of two parts, in the first part we generalise the integer Fisher's equation to be a space-time fractional differential equation and consider travelling wave solutions. In the second part we generalise an integer SIR model with spatial heterogeneity, which was studied by Murray [117], to a space-time fractional derivative model. We apply the (G0/G)-expansion method and find travelling wave solutions, although in this case we must consider the Jumarie's modified Riemann-Liouville fractional derivative. Finally, we consider the effect of changing the orders of time and space fractional derivatives on the location and speed of the travelling wave solution.This thesis considers the mathematical modelling of disease, using fractional differential equations in order to provide a tool for the description of memory effects. In Chapter 3 we illustrate a commensurate fractional order tumor model, and we find a critical value of the fractional derivative dependent on the parameter values of the model. For fractional derivatives of orders less than the critical value an unstable equilibrium point of the system becomes stable. In order to show changes in the observed areas of attraction of two stable points in the system, we then consider a fractional order SIR epidemic model and investigate the change from a monostable to a bistable system.;Chapter 4 considers a model for virus dynamics where the fractional orders for populations are different, called an incommensurate system. An approximate analytical solution for the characteristic equation of the incommensurate model is found when the different fractional orders are similar and close to the critical value of the fractional order of the commensurate system. In addition, the instability boundary is found as a function of both parameters. A comparison between analytical and numerical results shows the high accuracy of this approximation.;Chapter 5 consists of two parts, in the first part we generalise the integer Fisher's equation to be a space-time fractional differential equation and consider travelling wave solutions. In the second part we generalise an integer SIR model with spatial heterogeneity, which was studied by Murray [117], to a space-time fractional derivative model. We apply the (G0/G)-expansion method and find travelling wave solutions, although in this case we must consider the Jumarie's modified Riemann-Liouville fractional derivative. Finally, we consider the effect of changing the orders of time and space fractional derivatives on the location and speed of the travelling wave solution