Part I Linear Differential and Difference Equations: 1 Introduction to Linear Population Dynamics.- 2 Existence and Uniqueness of Solutions.- 3 Stability and Instability of Linear Systems.- 4 Positivity and Perron-Frobenius Theorem.- Part II NonLinear Differential: 5 Nonlinear Differential Equation.- 6 The Linearized Stability Principle and the Hartman-Grobman Theorem.- 7 Positivity and Invariant Sub-Regions.- 8 Monotone Semiflows.- Part III Applications to Epidemic Models: 9 Understanding and Predicting Unreported Cases in the 2019-nCov Epidemic Outbreak in Wuhan, China, and the Importance of Major Public Health Interventions.- 10 The COVID-19 Outbreak in Japan: Unreported Age-Dependent Cases.- 11 Clarifying Predictions for COVID-19 from Testing Data: The Example of New York State.- 12 SI Epidemic Model Applied to COVID-19 Data in Mainland China.- 13 A Robust Phenomenological Approach to Investigating COVID-19 Data for France.- 14 What Can We Learn From COVID-19 Data By Using Epidemic Models With Unidentified Infectious Cases?.- 15 Supplementary material.