Thanks for joining me!
Let us begin our journey through the amazing field of Biomathematics with a quote from a great scientist:
“Sometimes science is more art than science, a lot of people don’t get that.”
— Rick Sanchez
Although time and multiverse traveling has not been invented yet, we might be on the right path to find the cure or improve current treatments to deadly diseases such as cancer by relying on Biomathematics.
Biomathematics may be defined as the application of mathematical modelling in order to understand any kind of phenomena in biology, such as cancer, pharmacokinetics, diabetes, HIV, heart diseases, etc. Mathematical models may provide critical insights concerning the short and long-term evolution of complex biological dynamics.
BioMath focuses on the analysis and modelling of biological systems described by first order ordinary differential equations (ODEs). By combining the Localization of Compact Invariant Sets (LCCI) method with Lyapunov’s Stability Theory and LaSalle’s Invariance Principle, we can analyze the global dynamics of these biological systems. By means of the LCCI method we compute what we define as the localizing domain (a region on the state space where all compact invariant sets of a system are located).
Stability theories allows us to establish sufficient conditions for local or global stability of the largest compact invariant set, which may be an equilibrium point, a periodic orbit, a limit cycle or a chaotic attractor. Then, if all conditions are fulfilled, we ensure the existence of a Bounded Positively Invariant Domain (BPID), which, when we talk about mathematical models of biological systems, is usually located in the non-negative orthant.
In the case of cancer systems, we have been able to establish tumor clearance conditions, as we demonstrated in the paper: Global stability and tumor clearance conditions for a cancer chemotherapy system, which I kindly invite you to read here. Just to give you a glimpse, in the following picture we illustrate the global asymptotic stability of the tumor-free equilibrium point.
Dear reader, thank you for getting all the way here. In my Publications page you will find my postgraduate theses, journal papers, books and congress participations.