Mathematics for Life Science and Medicine (eBook)

Y.Takeuchi is a professor of Systems Engineering Department at Shizuoka University, Japan, where he has been on the faculty since 1979. He received his B.Eng.(1974), M.Eng.(1976) and Ph.D.(1979) at Kyoto University. In 1986, 1992, 1998, 2000 and 2002, Dr. Takeuchi was a visiting professor at Universita di Urbino, and in 1987-1988, at University of Alberta. Y. Iwasa is a professor of Department of Biology, Faculty of Sciences, Kyushu University, Japan, where he has been on the faculty since 1985. He received his B.S.(1975), M.Eng.(1977) and Ph.D.(1980) at Kyoto University. In 2003 and 2004, Dr. Iwasa was a visiting professor at Harvard University, and in 2002-2003 a member of Institute of Advanced Study, Princeton. K. Sato is an associate professor of Systems Engineering Department at Shizuoka University, Japan, where he has been on the faculty since 1996. He received his B.Sci.(1988) at University of Tsukuba, M.Sci.(1990) and Ph.D.(1993) at Kyushu University. From 1994 to 1996, Dr. Sato was a lecturer and an associate professor at Muroran Institute of Technology.

Preface 6
Contents 8
List of Contributors 9
1 Mathematical Studies of Dynamics and Evolution of Infectious Diseases 11
References 14
2 Basic Knowledge and Developing Tendencies in Epidemic Dynamics 15
2.1 Introduction 15
2.2 The fundamental forms and the basic concepts of epidemic models 16
2.3 Some tendencies in the development of epidemic dynamics 25
References 53
3 Delayed SIR Epidemic Models for Vector Diseases 60
3.1 Introduction 60
3.2 SIR epidemic models with constant birth rate and time delays 62
3.3 SIR epidemic models with time delays and non-constant birth rate 69
3.4 SIR epidemic models with time delays and non-constant birth rate 71
References 73
4 Epidemic Models with Population Dispersal 75
4.1 Introduction 75
4.2 Epidemic models with immigration of infectives 76
4.3 Constant population sizes in each patch 77
4.4 Multi-patches with demographic structure 82
4.5 A constant size with a standard incidence 90
4.6 Patch models with differentiating residence 94
4.7 Models with residence and demographic structure 96
4.8 Discussion 99
References 100
5 Spatial-Temporal Dynamics in Nonlocal Epidemiological Models 104
5.1 Introduction 104
5.2 Kermack–McKendrick model 106
5.3 Kendall model 107
5.4 Diekmann–Thieme model 110
5.5 Migration and spatial spread 113
5.6 A vector-disease model 116
5.7 Discusion 125
References 126
6 Pathogen Competition and Coexistence and the Evolution of Virulence 130
6.1 Introduction 130
6.2 Host populations with nonlinear birth rates and arbitrary incidence 132
6.3 Host populations with linear birth rates 137
6.4 Exponential growth and mass action incidence 139
6.5 Density-dependent per capita mortality and mass- action incidence 145
6.6 Linear birth rates and standard incidence 146
6.7 Discussion 152
6.8 Appendix 154
References 156
7 Directional Evolution of Virus Within a Host Under Immune Selection 161
7.1 Introduction 161
7.2 Model of cytotoxic immunity 163
7.3 Cytotoxic immunity with proportional activation term 166
7.4 Models of immune impairment 168
7.5 Proof of directional evolution 170
7.6 Target cells are helper T cells 173
7.7 General cross–immunity violates the fundamental theorem 174
7.8 Discussion 175
Appendix A 177
Appendix B 179
References 180
8 Stability Analysis of a Mathematical Model of the Immune Response with Delays 183
8.1 Introduction 183
8.2 Timing of innate and adaptive immunity 183
8.3 Analytical results 189
8.4 Characteristic equation and local stability 194
8.5 Numerical simulations 204
8.6 Discussion 208
8.7 Biological discussion 209
References 210
9 Modeling Cancer Treatment Using Competition: A Survey 213
9.1 Introduction 213
9.2 The no treatment case 214
9.3 Treatment by radiation 214
9.4 Treatment by chemotherapy 220
9.5 Treatment by immunotherapy 224
9.6 Metastasis 226
9.7 Discussion 229
References 229
Index 230