MATHEMATICAL MODELS APPLIED TO THE ENVIRONMENT

Knowing how to construct and understand mathematical models that describe qualitatively and quantitatively some phenomena related to the environment. Knowing how to use the main concepts of differential equation theory for application in the biological, geological and environmental fields. Knowing how to predict and justify the evolution of simple phenomena, described by ordinary differential equations, related to the biological, geological and environmental sciences.

In particular, the learning objectives of the course, according to the Dublin descriptors, are:

• Knowledge and understanding: The student will learn some basic concepts related to mathematical modelling and will develop both computing ability and the capacity of manipulating some common mathematical models.
• Applying knowledge and understanding: The student will be able to apply the acquired knowledge in the basic processes of mathematical modeling of classical problems arising from Biology and Environmental Sciences.
• Makin​g judgements: The student will be stimulated to autonomously deepen his/her knowledge. Constructive discussion between students and constant discussion with the teacher will be strongly recommended so that the student will be able to critically monitor his/her own learning process.
• Communication skills: The frequency of the lessons and the reading of the recommended books will help the student to be familiar with the rigor of the mathematical language. Through constant interaction with the teacher, the student will learn to communicate the acquired knowledge with rigor and clarity, both in oral and written form. At the end of the course the student will have learned that mathematical language is useful for describing in details some phenomena that arise from Applied Sciences.
• Learning skills: The student will be guided in the process of perfecting his/her study method. In particular, through suitable guided investigations, he/she will be able to independently tackle new topics, recognizing the necessary prerequisites to understand them.

1. Mathematical background. Numerical sets. Functions. Topology in R. Limits and continuity of real functions of one real variable. Summary of Differential Calculus and applications. Summary of Integral Calculus and applications. 
2. Elements of Statistics, Combinatorics and Discrete Probability. Data representations. Mean, median and fashion. Variance. Line of least squares. Interpolation techniques. Arrangements, permutations and combinations (simple and with repetition). Events and frequency. Classical and frequentist probability. Conditional probability. Stochastic dependence and compound probability. Bayes formula. Diagnostic texts.
3. Ordinary differential equations. Differential equations and physical models: definitions and terminology. First-order differential equations: separable differential equations and linear differential equations. Second-order differential equations.
4. Models in population dynamics. Malthus model and its generalizations. Verhulst models and its generalizations. Stability of the equilibrium solutions of Malthus and Verhulst models.
5. Models for environmental systems. Model for the evaluation of the quality of the green in the absence of residences. Model for the evaluation of the quality of green areas with residences. Model for the evaluation of biological energy production and diffusivity in an environmental system: introduction, ecological graph model, calculation of parameters, construction of the model and analysis.
6. Deterministic Models in Epidemiology. Epidemic SI model (construction and analysis), endemic Si model (construction and analysis), epidemic SIS model (construction and analysis), endemic SIS model (construction and analysis), epidemic SIR model (construction), endemic SIR model (construction), SEIR model (without details), MSEIR model (without details).

1. N. Hritonenko, Y. Yatsenko - Mathematical Modeling in Economics, Ecology and the Environment. Second edition – Springer (2013).
2. Lecture notes.