Global analysis of a cancer model with drug resistance due to Lamarckian induction and microvesicle transfer

Development of resistance to chemotherapy in cancer patients strongly effects the outcome of the treatment. Due to chemotherapeutic agents, resistance can emerge by Darwinian evolution. Besides this, acquired drug resistance may arise via changes in gene expression. A recent discovery in cancer research uncovered a third possibility, indicating that this phenotype conversion can occur through the transfer of microvesicles from resistant to sensitive cells, a mechanism resembling the spread of an infectious agent. We present a model describing the evolution of sensitive and resistant tumour cells considering Darwinian selection, Lamarckian induction and microvesicle transfer. We identify three threshold parameters which determine the existence and stability of the three possible equilibria. Using a simple Dulac function, we give a complete description of the dynamics of the model depending on the three threshold parameters. We demonstrate the possible effects of increasing drug concentration, and characterize the possible bifurcation sequences. Our results show that the presence of microvesicle transfer cannot ruin a therapy that otherwise leads to extinction, however it may doom a partially successful therapy to failure.

Elemental access to limit cycle existence in Biomath education.

This paper originated from the desire to develop elementary calculus based tools to empower students, not necessarily with a strong mathematical background, to test predator-prey related models for boundedness of solutions and for the existence of limit cycles. There are several well-known methods available to prove, or disprove, the existence of bounded solutions to systems of differential equations. These methods rely on Liénard's theorem or using Dulac or Lyaponov functions. The level of mathematics required in the study of differential equations is not addressed in the courses presented on the first year level, and students in biology, ecology, economics and other fields are often not suitably equipped to perform these advanced techniques.The conditions under which a unique limit cycle exists in predator-prey systems is considered a primary problem in mathematical ecology. A great deal of mathematical effort has gone into trying to establish simple, yet general, theorems which will allow one to decide whether a given set of nonlinear equations has a limit cycle or not. We introduce a method to first determine the boundedness of solution trajectories in such a way that the transformation to a Liénard system or the use of a Dulac function can be avoided. Once boundedness of trajectories has been established, the nature of the equilibrium points reduces to simple eigenvalue analysis. The Elemental Limit Cycle method (ELC) provides elementary criteria to evaluate the nature of the pivotal functions of a system which will indicate boundedness and may be applicable to more general models.

Construction of Dulac functions for mathematical models in population biology

In this paper, we present a method for constructing a Dulac function for mathematical models in population biology, in the form of systems of ordinary differential equations in the plane.

Dynamic Analysis of an SIRS Model with Nonlinear Incidence Rate

In this paper, we study the global dynamics of an SIRS epidemic model with nonlinear inci- dence rate. By means of Dulac function and Poincare-Bendixson Theorem, we proved the global asy- mptotical stable results of the disease-free equilibrium. It is then obtained the model undergoes Hopf bifurcation and existence of one limit cycle. Some numerical simulations are given to illustrate the an- alytical results.

Dynamic Behavior for an SIRS Model with Nonlinear Incidence Rate

An SIRS epidemic model with nonlinear incidence rate is studied. It is assumed that susceptible and infectious individuals have constant immigration rates. By means of Dulac function and Poincare-Bendixson Theorem, we proved the global asymptotical stable results of the disease-free equilibrium. It is then obtained the model undergoes Hopf bifurcation and existence of one limit cycle. Some numerical simulations are given to illustrate the analytical results.