Mathematics applied to biology

This course aims to provide students with an update on basic infinitesimal calculus along with an initial approach to modeling and solving some problems in ordinary differential equations related to biology. Students, whether familiar or not with the usual simple techniques of calculus, will expand their knowledge and discover it as an indispensable tool for mathematically formalizing and solving many problems that arise in the field of biology.

LECTURE CLASSES (36 hours)

TOPIC 1: DISCRETE BIOLOGICAL MODELS WITH MATRICES (5 hours)
Matrices and matrix calculus. Eigenvalues and eigenvectors of a matrix. Introduction to modeling an age-structured population in discrete time: Leslie-Lewis matrix.

TOPIC 2: IMPORTANCE OF FUNCTIONS IN THE FIELD OF BIOLOGY (16 hours)
Exponential growth and decay. Logarithmic function: logarithmic and semi-logarithmic graphs. Level curves. Interpretation of the Shannon diversity index. Rate of change for a variable. Monod kinetic model. Rates of change for two variables. Gradient vector field: chemotaxis. Holling's disc equation. Reversing differentiation: calculation of antiderivatives.

TOPIC 3: BIOLOGICAL MODELS WITH DIFFERENTIAL EQUATIONS (15 hours)
From the rate of change as a hypothesis to the differential equation: introduction to modeling a biological process using differential equations. Solving differential equations. Biological models with differential equations: Malthusian law, logistic equation, and von Bertalanffy equation. Biological models with systems of linear differential equations: compartment models. Biological models with systems of nonlinear differential equations: the Lotka-Volterra predator-prey model.

SEMINAR CLASSES (8 hours)
Exercises on discrete biological models with matrices. (1 hour)
Exercises on functions in the field of biology. (4 hours)
Exercises on biological models with differential equations. (3 hours)


INTERACTIVE CLASSES IN THE COMPUTER LAB (9 hours)
- Computer practices:
Practice 1 (2 hours). Introduction to the mathematical software environment MATLAB. Matrix calculus. Computation of the eigenvalues and the eigenvectors of a matrix.
Practice 2 (2 hours). Introduction to symbolic computation with MATLAB. Elementary mathematical functions and graphical representations with MATLAB.
Practice 3 (1 hour). Differentiation and integration with MATLAB.
Practice 4 (1 hour). Solving differential equations with MATLAB (I).
Practice 5 (1 hour). Solving differential equations with MATLAB (II).
- Evaluation of the contents of the computer practices (2 hours).

TUTORING (1 hour)
Clarifying any doubts that may arise regarding the subject, both from a conceptual standpoint and regarding the computer tool used.

EXAM (3 hours)

BASIC BIBLIOGRAPHY
[1] Neuhauser, C. (2004) Matemáticas para ciencias. 2a ed. Madrid: Pearson Prentice Hall. Available online.
[2] Pérez López, C. (2011) MATLAB a través de ejemplos. Madrid: Ibergarceta.
[3] Zill, D. G. (2015) Ecuaciones diferenciales con aplicaciones de modelado. Décima edición. México, D.F: Cengage Learning. Available online.

COMPLEMENTARY BIBLIOGRAPHY
[1] Braun, M. (1990) Ecuaciones diferenciales y sus aplicaciones. México: Grupo Editorial Iberoamérica.
[2] Derrick, W. R. & Grossman, S. I. (1997) Elementary differential equations. 4th ed. Reading, Massachusetts: Addison-Wesley.
[3] Edwards, C. H. & Penney, D. E. (1994) Ecuaciones diferenciales elementales y problemas con condiciones en la frontera. 3a ed. México: Prentice-Hall Hispanoamericana.
[4] Nagle, R. K. & Saff, E. B. (1992) Fundamentos de ecuaciones diferenciales. 2a ed. Argentina: Addison-Wesley Iberoamericana.
[5] Simmons, G. F. (1999) Ecuaciones diferenciales con aplicaciones y notas históricas. 2a ed. Madrid: McGraw-Hill.
[6] Stein, S. K. & Barcellos, A. (1995) Cálculo y geometría analítica. 5a ed. Santafé de Bogotá: McGraw-Hill.

As indicated in the report of the Bachelor's Degree in Biology's, third edition, this course has the following learning outcomes:
- Knowledge/Contents: Con01, Con04, Con10.
- Skills/Abilities: H/D01, H/D02, H/D03, H/D04, H/D07, H/D08, H/D09, H/D014.
- Competencies: Comp01, Comp02, Comp03, Comp05, Comp07.

The following teaching methodologies listed in the third edition's memory of the Bachelor's Degree in Biology will be used:

- Lectures.
- Problem-solving sessions.
- Seminars.
- Tutoring.
- Computer Lab Practices.
- E-learning.

For the computation of the final grade (FG) in the first attempt, the grades of continuous assessment tests and the final exam will be considered according to the following criteria:

- The continuous assessment grade will carry a weight of 30% in the FG and will be based on the completion of no more than three tests, which will be conducted within the scheduled class time for the subject. Both the content covered in seminars or lectures and that of computer lab practices will be examined in these tests.

- The final exam will carry a weight of 70% in the FG.

Regarding the final grade for the second attempt, the grade obtained in the final exam of the first attempt will be replaced by the corresponding grade obtained in the second attempt.

The evaluation system for repeating students is the same as described above for first-time enrollees. No grades obtained in previous courses will be retained.

The aforementioned instruments will be used to evaluate the knowledge/contents, skills/abilities, and competencies of the subject, according to the following scheme:
- In the continuous assessment tests not related to computer practices and in the final exam: Con01, Con04, Con10, H/D02, H/D04, H/D08, Comp01, Comp03, Comp05.
- In the evaluation of computer lab practices: Con01, Con04, Con10, H/D01, H/D02, H/D03, H/D04, H/D07, H/D08, H/D09, H/D14, Comp01, Comp02, Comp03, Comp05, Comp07.

Lecture classes: 36.
Interactive seminar classes: 8.
Interactive classes in the computer lab: 9.
Tutoring sessions: 1.
Conducting exams: 3.
Total hours of contact with the instructor: 57.

Individual study: 46.
Recommended readings, library activities, or similar: 5.
Personal work for computer lab practices: 15.
Preparation of assignments, exercises, etc.: 25.
Attendance to talks or other recommended activities: 2.
Total hours of personal study: 93.

- Daily review of the topics covered in class.
- Solving the exercises proposed in the exercise booklets.
- Utilizing professors' tutoring hours to address any doubts regarding the subject matter.

- Expository and interactive teaching will be conducted in person and complemented by the virtual course of the subject on the Moodle platform, where students will find materials related to the contents covered in different sessions, including exercise booklets and various bibliographic materials.

- Tutoring sessions will be conducted in person or via university electronic tools such as email or MS Teams.

- Cases of fraudulent completion of exercises or tests will be handled in accordance with the regulations outlined in the USC's guidelines ("Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións").