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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.
REMARKS: NOTE: the authors have supplied variants on the promotion text that are more suitable for promotionin different fields (by virtue of different emphasis in the content). They are not enclosed, but in the mathematics editorial.
| 1 | Introduction | 1 |
| 1.1 | What is Vertical Transmission? | 1 |
| 1.2 | Methodology, Terminology and Notation | 2 |
| 1.3 | Examples of Vertically Transmitted Diseases | 5 |
| 1.4 | Organization and Principal Results | 7 |
| 2 | Differential Eguations Models | 11 |
| 2.1 | A Classical Model Extended | 11 |
| 2.2 | Some Biological and Modeling Considerations | 15 |
| 2.3 | Model Without Immune Class | 17 |
| 2.4 | Discussion of the Global Result | 22 |
| 2.5 | Proofs of the Results | 30 |
| 2.6 | No Horizontal Transmission | 32 |
| 2.7 | The Model with Immune Class | 34 |
| 2.7.1 | The SIR Model with Vaccination | 37 |
| 2.8 | The Case of Constant Population | 38 |
| 2.9 | A Model with Vaccination | 39 |
| 2.10 | Models with Latency or Maturation Time | 41 |
| 2.11 | Models with Density Dependent Death Rate | 44 |
| 2.12 | Parameter Estimation | 46 |
| 2.13 | Models of Chagas' Disease | 49 |
| 2.13.1 | Proportional Mixing and Vector Transmission | 49 |
| 2.13.2 | An SIS Model with Proportional Mixing | 56 |
| 2.13.3 | Logitic Control | 59 |
| 2.14 | An SIRS Model with Proportional Mixing | 61 |
| 2.15 | Evolution of Viruses | 64 |
| 2.16 | The Mathemathial Background | 66 |
| 2.16.1 | Positivity and Invariant Regions | 67 |
| 2.16.2 | Equilibria and Stability Analysis | 68 |
| 2.16.3 | Global Stability in One and Two Dimensions | 71 |
| 2.16.4 | General Global Stability | 73 |
| 2.16.5 | A Special Global Stability Result | 74 |
| 2.16.6 | Existence and Bifurcation of Periodic Solutions | 76 |
| 3 | Difference Equations Models | 79 |
| 3.1 | Introduction | 79 |
| 3.2 | A Model for the Transmission of Keystone Virus | 80 |
| 3.3 | Population Size Control via Vertical Transmission | 84 |
| 3.3.1 | Fine Structure of Population Size Control | 90 |
| 3.3.2 | Proofs of the Theorems | 94 |
| 3.4 | Vertical Transmission in Insect Populations | 106 |
| 3.5 | Logistic Control in the Reproduction Rate | 114 |
| 3.5.1 | Complicated Dynamics and Long Term Transients | 122 |
| 3.6 | Logistic Control through the Death Terms | 129 |
| 3.6.1 | Synchronous Oviposition | 129 |
| 3.6.2 | Distributed Asynchronous Oviposition | 134 |
| 3.7 | Mathematical Background | 135 |
| 3.7.1 | Positivity and Invariant Regions | 135 |
| 3.7.2 | Equilibria and Stability Analysis | 136 |
| 3.7.3 | Global Stability | 138 |
| 3.7.4 | Periodic Solutions, Bifurcation and Chaos | 139 |
| 4 | Delay Differential Equations Models | 143 |
| 4.1 | The Role of Delays in Epidemic Models | 143 |
| 4.2 | A Model with Maturation Delays | 144 |
| 4.3 | Delays Due to Partial Immunity | 148 |
| 4.4 | Delay Due to an Incubation Period | 151 |
| 4.5 | A Model with Spatial Diffusion | 154 |
| 4.6 | Diseases with Long Subclinical Periods | 156 |
| 4.7 | Mathematical Background | 159 |
| 4.7.1 | Positivity and Invariant Regions | 160 |
| 4.7.2 | Equilibria and Stability Analysis | 161 |
| 4.7.3 | Liapunov Stability Theory | 165 |
| 4.7.4 | Existence and Bifurcation of Periodic Solutions | 167 |
| 4.7.5 | Invariant Integral Conditions | 168 |
| 5 | Age and Internal Structure | 171 |
| 5.1 | Age Structure and Vertical Transmission | 171 |
| 5.2 | Modeling Internal Structure | 173 |
| 5.3 | Derivation of the Model Equations | 176 |
| 5.4 | Age Structure and the Catalytic Curve | 179 |
| 5.5 | An s[actual symbol not reproducible]i Model with Vertical Transmission | 184 |
| 5.6 | Analysis of the Intracohort Model | 193 |
| 5.7 | Analysis of the Intercohort s[actual symbol not reproducible]i[actual symbol not reproducible]s Model | 198 |
| 5.8 | Numerical Simulations | 201 |
| 5.9 | Global Behavior of the s[actual symbol not reproducible]i[actual symbol not reproducible]s Model | 206 |
| 5.10 | Destabilization Due to Age Structure | 215 |
| 5.11 | Thresholds in Age Dependent Models | 217 |
| 5.12 | Spatial Structure | 222 |
| 5.13 | The Force of Infection Terms | 225 |
| References | 231 | |
| Author Index | 241 | |
| Subject Index | 245 |
Overview
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 ...