Estimation of the infection rate in epidemic models with multiple populations

CHAPTER 1 Introduction and motivation Throughout history, infectious diseases have profoundly aected human development: for example, the Black Death (bubonic plague) that swept Europe in several waves during the fourteenth century is estimated to have caused the death of as much as one-third of the European population, re- curring regularly for more than three centuries [9]. The defeat of the Aztecs and Incas by invading Spaniards in 1519 and 1532, respectively, can also be partially attributed to outbreaks of infectious diseases, such as smallpox, measles and diphtheria, that were imported from Europe and to which the invaders were mostly immune. It is estimated that the population of Mex- ico was reduced to one-tenth of its previous size between 1519 and 1530 [9]. Further examples can be found in the book by McNeill [34], to which the reader interested in the history of epidemics is referred. In view of the im- portance of infectious diseases, people naturally started investigating their causes and searching for treatments; one of the oldest accounts is the book by Hippocrates [25] (see also [31]), who is often referred to as the father of Western medicine. The existence of microorganisms was not discovered be- fore the seventeenth century, with the aid of the rst microscopes; however, their role in the spread of infectious diseases was understood much later [22]: a rst theory of infectious agents (also called pathogens) was developed only in the latter part of the nineteenth century [9]. The rst application of mathematical modelling to infectious diseases is due to Bernoulli [5] (see also [6]), who developed models to understand the eectiveness of a mass vaccination campaign against smallpox in increasing life expectancy. The foundations of the mathematical epidemiology of in- fectious diseases were laid only about a century later by En’ko [17] (see also [15, 18]), Ross [38], and Kermack and McKendrick [27, 28, 29]; as we shall see, some of their ideas, such as that of a threshold behaviour, can still be found in more recent models. The aim of mathematical epidemiology of infectious diseases is threefold [13, Ch. 1]: (a) the rst objective is to understand the biological and so- cial mechanisms of disease spread by means of an appropriate mathematical structure that models available data; (b) the second goal is prediction of future epidemics, including assessment of the possible impact of outbreaks and estimation of associated medical costs; (c) the third aim is to under- stand how the spread may be controlled, for example through education, immunization and isolation, by introducing these measures in the model 1