Full Description
Infectious diseases are transmitted through various
different mechanisms including person to person
interactions, by insect vectors and via vertical
transmission from a parent to an unborn offspring. The
population dynamics of such disease transmission can be very
complicated and the development of rational strategies for
controlling and preventing the spread of these diseases
requires careful modeling and analysis.
The book describes current methods for formulating models
and analyzing the dynamics of the propagation of diseases
which include vertical transmission as one of the mechanisms
for their spread. Generic models that describe broad classes
of diseases as well as models that are tailored to the
dynamics of a specific infection are formulated and
analyzed. The effects of incubation periods, maturation
delays, and age-structure, interactions between disease
transmission and demographic changes, population crowding,
spatial spread, chaotic dynamic behavior, seasonal
periodicities and discrete time interval events are studied
within the context of specific disease transmission models.
No previous background in disease transmission modeling and
analysis is assumedand the required biological concepts and
mathematical methods are gradually introduced within the
context of specific disease transmission models. Graphs are
widely used to illustrate and explain the modeling
assumptions and results.
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Contents
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.
