1 Introduction.- 1.1 What is Vertical Transmission?.- 1.2 Methodology, Terminology and Notation.- 1.3 Examples of Vertically Transmitted Diseases.- 1.4 Organization and Principal Results.- 2 Differential Equations Models.- 2.1 A Classical Model Extended.- 2.2 Some Biological and Modeling Considerations.- 2.3 Model Without Immune Class.- 2.4 Discussion of the Global Result.- 2.5 Proofs of the Results.- 2.6 No Horizontal Transmission.- 2.7 The Model with Immune Class.- 2.8 The Case of Constant Population.- 2.9 A Model with Vaccination.- 2.10 Models with Latency or Maturation Time.- 2.11 Models with Density Dependent Death Rate.- 2.12 Parameter Estimation.- 2.13 Models of Chagas’ Disease.- 2.14 An SIRS Model with Proportional Mixing.- 2.15 Evolution of Viruses.- 2.16 The Mathematical Background.- 3 Difference Equations Models.- 3.1 Introduction.- 3.2 A Model for the Transmission of Keystone Virus.- 3.3 Population Size Control via Vertical Transmission.- 3.4 Vertical Transmission in Insect Populations.- 3.5 Logistic Control in the Reproduction Rate.- 3.6 Logistic Control through the Death Terms.- 3.7 Mathematical Background.- 4 Delay Differential Equations Models.- 4.1 The Role of Delays in Epidemic Models.- 4.2 A Model with Maturation Delays.- 4.3 Delays Due to Partial Immunity.- 4.4 Delay Due to an Incubation Period.- 4.5 A Model with Spatial Diffusion.- 4.6 Diseases with Long Subclinical Periods.- 4.7 Mathematical Background.- 5 Age and Internal Structure.- 5.1 Age Structure and Vertical Transmission.- 5.2 Modeling Internal Structure.- 5.3 Derivation of the Model Equations.- 5.4 Age Structure and the Catalytic Curve.- 5.5 An s ? i Model with Vertical Transmission.- 5.6 Analysis of the Intracohort Model.- 5.7 Analysis of the Intercohort s ? i ? s Model.- 5.8Numerical Simulations.- 5.9 Global Behavior of the s ? i ? s Model.- 5.10 Destabilization Due to Age Structure.- 5.11 Thresholds in Age Dependent Models.- 5.12 Spatial Structure.- 5.13 The Force of Infection Terms.- References.- Author Index.