Computational Algebraic Geometry is applied to the analysis of various epidemic models for Schistosomiasis and Dengue, both, for the case without control measures and for the case where control measures are applied. The models were analyzed using the mathematical software Maple. Explicitly the analysis is performed using Groebner basis, Hilbert dimension and Hilbert polynomials. These computational tools are included automatically in Maple. Each of these models is represented by a system of ordinary differential equations, and for each model the basic reproductive number (R0) is calculated. The effects of the control measures are observed by the changes in the algebraic structure of R0, the changes in Groebner basis, the changes in Hilbert dimension, and the changes in Hilbert polynomials. It is hoped that the results obtained in this paper become of importance for designing control measures against the epidemic diseases described. For future researches it is proposed the use of algebraic epidemiology to analyze models for airborne and waterborne diseases.
A food freezing model is analyzed analytically. The model is based on the heat diffusion equation in the case
of cylindrical shaped food frozen by liquid nitrogen; and assuming that the thermal conductivity of the
cylindrical food is radially modulated. The model is solved using the Laplace transform method, the
Bromwich theorem, and the residue theorem. The temperature profile in the cylindrical food is presented as
an infinite series of special functions. All the required computations are performed with computer algebra
software, specifically Maple. Using the numeric values of the thermal and geometric parameters for the
cylindrical food, as well as the thermal parameters of the liquid nitrogen freezing system, the temporal
evolution of the temperature in different regions in the interior of the cylindrical food is presented both
analytically and graphically. The duration of the liquid nitrogen freezing process to achieve the specified
effect on the cylindrical food is computed. The analytical results are expected to be of importance in food
engineering and cooking engineering. As a future research line, the formulation and solution of freezing
models with thermal memory is proposed.
Toy models for the Arabidopsis Thaliana flower and the Drosophila are analyzed using Microsoft SMT-Solver Z3 with
the SMT-LIB language. The models are formulated as Boolean networks which describe the metabolic cycles for
Arabidopsis and Drosophila. The dynamic activation of the different bio macromolecules is described by the variables
and laws of Boolean transition. Specifically, bitvectors and assertions, which describe the change of state of bitvectors
from a sampling time to the next, are used. The dynamic feasibility problem of the biological network is translated to a
Boolean satisfiability problem. The corresponding dynamic attractors are represented as a model of satisfiability. The Z3
software allows all required computations in a friendly and efficient manner. It is expected that the SMT-solvers, such as
Z3, will become a routine tool in system biology and that they will provide bio-nanosystem design techniques. As a line
for future research, the study of the models for Arabidopsis and Drosophila using different SMT-solvers such as CVC4,
Mathsat and Yices, is proposed.