# Mathematical Models in Infectious Disease Epidemiology

Chapter
Part of the Statistics for Biology and Health book series (SBH)

## Abstract

The idea that transmission and spread of infectious diseases follows laws that can be formulated in mathematical language is old. In 1766 Daniel Bernoulli published an article where he described the effects of smallpox variolation (a precursor of vaccination) on life expectancy using mathematical life table analysis (Dietz and Heesterbeek 2000). However, it was only in the twentieth century that the nonlinear dynamics of infectious disease transmission was really understood. In the beginning of that century there was much discussion about why an epidemic ended before all susceptibles were infected with hypotheses about changing virulence of the pathogen during the epidemic.

## Keywords

Vaccination Coverage Reproduction Number Epidemic Modeling Infected Person Susceptible Population
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Andersson H, Britton T (2000) Stochastic epidemic models and their statistical analysis. New York: Springer
2. Anderson RM, May RM (1991) Infectious disease of humans: dynamics and control. Oxford: Oxford University PressGoogle Scholar
3. Bailey NTG (1975) The mathematical theory of infectious diseases and its applications. 2nd ed. London: Griffin
4. Becker NG (1989) Analysis of infectious disease data. London: Chapman and HallGoogle Scholar
5. Cauchemez S, Boelle PY, Thomas G, Valleron AJ (2006) Estimating in real time the efficacy of measures to control emerging communicable diseases. Am J Epidemiol; 164(6):591–7
6. Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases. Chichester: WileyGoogle Scholar
7. Diekmann O, Heesterbeek JA, Metz JA (1990) On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J Math Biol; 28(4):365–82
8. Dietz K, Heesterbeek JA (2000) Bernoulli was ahead of modern epidemiology. Nature; 408(6812):513–4
9. Eames KT, Keeling MJ (2003) Contact tracing and disease control. Proc Biol Sci; 270(1533):2565–71
10. Edmunds WJ, Medley GF, Nokes DJ (1999) Evaluating the cost-effectiveness of vaccination programmes: a dynamic perspective. Stat Med; 18(23):3263–82
11. Ferguson NM, Cummings DA, Fraser C, Cajka JC, Cooley PC, Burke DS (2006) Strategies for mitigating an influenza pandemic. Nature; 442(7101):448–52
12. Ferguson NM, Keeling MJ, Edmunds WJ, Gani R, Grenfell BT, Anderson RM, et al. (2003) Planning for smallpox outbreaks. Nature; 425(6959):681–5
13. Grundmann H, Hellriegel B (2006) Mathematical modelling: a tool for hospital infection control. Lancet Infect Dis; 6(1):39–45
14. Hadeler KP, Waldstatter R, Worz-Busekros A (1988) Models for pair formation in bisexual populations. J Math Biol; 26(6):635–49
15. Hamer WH (1906) Epidemic disease in England – the evidence of variability and persistency of type. Lancet; 1:733–39Google Scholar
16. Hethcote HW, Yorke JA (1984) Gonorrhea transmission dynamics and control. New York: Springer Verlag
17. Keeling MJ, Eames KT (2005) Networks and epidemic models. J R Soc Interface; 2(4):295–307
18. Keeling MJ, Rohani P (2007) Modeling infectious diseases in humans and animals. Princeton: Princeton University PressGoogle Scholar
19. Kermack WO, McKendrick AG (1991a) Contributions to the mathematical theory of epidemics–II. The problem of endemicity.1932. Bull Math Biol; 53(1–2):57–87Google Scholar
20. Kermack WO, McKendrick AG (1991b) Contributions to the mathematical theory of epidemics – I. 1927. Bull Math Biol; 53(1–2):33–55Google Scholar
21. Kermack WO, McKendrick AG (1991c) Contributions to the mathematical theory of epidemics–III. Further studies of the problem of endemicity.1933. Bull Math Biol; 53(1–2):89–118Google Scholar
22. Koopman J, Simon C, Jacquez J, Joseph J, Sattenspiel L, Park T (1988) Sexual partner selectiveness effects on homosexual HIV transmission dynamics. J Acquir Immune Defic Syndr; 1(5):486–504Google Scholar
23. Kretzschmar M, Welte R, van den Hoek A, Postma MJ (2001) Comparative model-based analysis of screening programs for Chlamydia trachomatis infections. Am J Epidemiol; 153(1):90–101
24. Liljeros F, Edling CR, Amaral LA, Stanley HE, Aberg Y (2001) The web of human sexual contacts. Nature; 411(6840):907–8
25. Meyers LA, Newman ME, Martin M, Schrag S (2003) Applying network theory to epidemics: control measures for Mycoplasma pneumoniae outbreaks. Emerg Infect Dis; 9(2):204–10Google Scholar
26. Mills CE, Robins JM, Lipsitch M (2004) Transmissibility of 1918 pandemic influenza. Nature; 432(7019):904–6
27. Peltola H, Davidkin I, Valle M, Paunio M, Hovi T, Heinonen OP, et al. (1997) No measles in Finland. Lancet; 350(9088):1364–5
28. Rohani P, Earn DJ, Grenfell BT (1999) Opposite patterns of synchrony in sympatric disease metapopulations. Science; 286(5441):968–71
29. Watts DJ, Strogatz SH (1998) Collective dynamics of 'small-world' networks. Nature; 393(6684):440–2
30. Wallinga J, Teunis P (2004) Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. Am J Epidemiol; 160(6):509–16