\chapter{Preface, Creed and Apology}

\section{What it is all about and what not}

Epidemiology is the study of the spread of diseases, in space and time, with
the objective to trace \textit{factors} that are responsible for, or
contribute to, their occurrence.

In 1854, John Snow demonstrated that cholera could be transmitted via
drinking water. He combined incidence data from an area surrounding
Broad Street in London with a sketch of the location of water pumps, and
noticed that the cases were clustered around a particular pump. This is a
classic example of how \textit{description} and \textit{data analysis} may
lead to an \textit{explanation}, which then can be used for \textit{%
prevention} (or \textit{prediction}). The general usefulness of maps in this
context is brilliantly illustrated in the \textit{Altas of Disease
Distributions}, by Cliff  \& Haggett (1988).

To achieve an understanding of complex phenomena, one needs many tools. This
book is not about maps and not about statistics (for the latter we refer to
Becker, 1989). It is about caricatural mathematical descriptions of the
mechanisms of transmission of infectious agents and about the analysis of
the models that result from such descriptions. It is about \textit{%
translating} assumptions concerning biological (behavioural, immunological,
demographical, medical) aspects into mathematics, about mathematical \textit{%
analysis} of certain classes of equations aided by interpretation, and about
the \textit{drawing of conclusions} where results from the mathematical
analysis are translated back into biology.

Another way of phrasing this is to say that this book is about \textit{%
thought experiments} that help to create conceptual clarity, to expose
hidden working hypotheses and to find mechanistic links between different
observable quantities. It tries to unravel the relationship between assumed
mechanisms at the individual level and the resulting phenomena at the
population level.

\section{The top down approach}

It is the mathematicians' inclination to strive for an abstract and general
theory, the hope being that, once such a theory exists, one can make it
operational by mere specification and elaboration. We are aware that this
view is too rosy. To apply general qualitative theory to the concrete and
the specific in order to generate quantitative conclusions is a highly
non-trivial affair requiring its own type of ingenuity. Top down does not
make bottom up redundant. It is an art to discern pattern in facts and
figures concerning a specific disease agent affecting a specific host
population.

We believe that collaboration between specialists in different fields is the
designated way to make top down and bottom up meet. This believe was
triggered by our own experience in working, sometimes indirectly, together
with experts from many areas of human, animal (both wild and farm) and plant
epidemiology. We realise that this book only offers the top down part of the
story. It provides \textit{tools} (concepts, methods, techniques, results)
and \textit{qualitative insights} (e.g. the Threshold Theorem of Section
1.2.1 and the asymptotic speed of propagation of Chapter 8) but does not
embark upon, for example, a quantitative study of the advantages and disadvantages
of various vaccination strategies against rubella in the Netherlands, or the
question of how many hospital beds are likely to be needed for AIDS patients in
the next twenty years.

However, we do claim that, for a mathematically oriented book, we are
exceedingly conscientious about the meaning and interpretation of the
assumptions that are made. Even though most of the models discussed are
deterministic, we think in terms of individuals and their behaviour, and
hence `deterministic' just means that there are assumed to be very many such
individuals and that, as a consequence, chance fluctuations are of
relatively minor importance. We strongly advocate the formulation of model
assumptions in terms of behaviour of individuals, and in this respect the
book is written with missionary zeal (even top experts have been known to
sometimes misinterpret quadratic/mass-action terms by jumping too recklessly
from the individual to the population level and back). Our hope is that
through formulation of population models in terms of behaviour of
individuals one can facilitate the linking up of the top down with the
bottom up approach.

\section{A workbook}

Some books offer wisdom. They can be read at leisure in an armchair near a
fireplace, provided one pauses every now and then for contemplation. This
is not such a book.

This book has a zillion exercises and begs to be read with pencil and paper
at hand (or, perhaps, in a more modern way, using a computer with some
program for symbolic manipulation). Some of the exercises you may just want
to read to see what statements they concern. This reading is essential, since
usually the exercises are an integrated part of the exposition. For many
exercises, however, mere reading is not enough: one actually needs to do them. Learning to
translate, model, analyse and interpret involves
training. Some exercises are ridiculously simple since we have tried not to
omit arguments or to tire the reader with details; where other writers
would state `one easily sees' or `as a simple argument shows', etc., we have
inserted an exercise. Other exercises, however, are very hard and elaborate.
We anticipate that our readers will feel at times frustrated or even
irritated. We therefore provide complete elaborations of all exercises, even
of the `ridiculously simple ones', as an integrated part of the book. When
you cannot do a specific exercise, we advise that you initially only briefly
glance at the elaboration as a kind of hint and then try again. Only if
all glances fail to inspire should you study the elaboration in detail.

We are not sadists who like to pester their readers with exercises. We are
convinced that the reward is enormous. In literally working through this
book the reader acquires modelling skills that are also valuable outside of
epidemiology, certainly within population dynamics, but even beyond that.
The reader receives training in mathematical argumentation, modelling and
analysis.

The book is primarily aimed at self-study. To study the book in a small
group that meets for discussion sessions, however, can be more stimulating
and rewarding (if only because by dividing the labour it is easier to keep
the spirits high). Finally, we trust that a lecturer can use the book as a
basis for a course on epidemic modelling. We invite readers to send us their
comments and constructive criticism.

\section{Portrait of the reader as a young person}

What is the audience that we have in mind? The answer depends on our temper.
In optimistic moods we expect that anybody with an interest in epidemic
modelling should be able to digest most of the text and benefit from it.
When realism strikes, we appreciate that a certain background in mathematics
is required, and we narrow down the description to applied mathematicians
(to-be) with an interest in population biology and epidemiology and to
theoretical biologists and epidemiologists with a strong inclination to
persevere when mathematical language at first seems to complicate, rather
than simplify, the modelling.

Our hope is that the applied mathematicians learn to see i) the subtleties of
model assumptions; ii) that continuous-time models not necessarily take the
form of a system of ordinary differential equations (`ODE') and iii) that
often biological interpretation suggests how to proceed with the
mathematical analysis.

Our hope is that the theoretical biologists and epidemiologists i) enlarge
their tool kit considerably and ii) conclude that sometimes abstraction may
actually make things simpler and more transparent.

Our ideal reader feels attracted by these educational aims.

\section{A brief outline of the book}

This book is divided into three parts and twelve chapters. In Part I, we
shall introduce the key questions, basic ideas, fundamental concepts and
mathematical arguments in as simple a context as possible. This entails in
particular that we treat all host individuals as identical with respect to
behaviour and physiology and that we deal with such concepts as thresholds,
final sizes for epidemics, repeated outbreaks, the endemic state and
population regulation.

When the host population is heterogeneous, we need more advanced
mathematics. To describe the initial phase of epidemic spread, we can
restrict attention to linear mathematics and a systematic approach is
possible. The theory, with many examples, is presented in Part II. In
addition we pay some (but not much) attention to nonlinear aspects in a
general setting. We shall pay some more attention to age structure and
spatial structure in separate chapters, since these are particularly relevant
for understanding of the population dynamics of many infective agents. To
analyse nonlinear structured models one is often forced to make debatable
simplifying assumptions. Even then, one needs to resort to tricks, for lack
of a powerful general theory. We therefore do not forage deeply into
nonlinear theory. For most of the examples in the book we have those
infective agents in mind that are usually collectively called
`microparasites', but in Chapter 9 we briefly touch upon some aspects where
`macroparasites' differ from `microparasites' (and where they do not), and
concentrate on the consequences that these differences and agreements have
for the mathematical treatment of invasion. In the final chapter of Part II
we pay attention to one of the fundamental and conceptually most difficult
aspects of epidemic theory: the myriad ways in which one can model contacts
between individuals.

Part III consists of two chapters. As a consequence of our educational aim,
we provide, in Chapter 11 and 12, complete elaborations of all exercises.
These elaborations are detailed and sometimes lengthy, and in this way often
serve as an extension and deepening of the main text. This makes the
elaborations into an integrated part of the book.

A final remark concerns our way of referring to the literature. The
literature of epidemic theory is extensive and growing steadily. It would be
very difficult, bordering on the impossible, to do justice to all valuable
contributions to the literature. We have deliberately chosen to write a textbook and not a review
of the state-of-the-art in epidemic theory. As a
consequence we have two types of references: local specialist literature
(mostly papers) and global general texts for further reading (mostly books).
The local literature is included in places where it is necessary for the
exposition at that point and is given in footnotes. The global references
are given in a short list near the end of the book. They are ordered
thematically and include background mathematics books. We do not usually
refer to our own papers, since much of the relevant material is represented
somehow in the text. The research was performed over a ten-year period with
many collaborators. Apart from the people mentioned explicitly in the next
section, the main collaborators whose efforts are presented here are Mick Roberts and Klaus
Dietz.

\section{And what about reality?}

So far our aim has been to position the book in a scientific landscape of
which we provided some topography. Here are a few final remarks in this
direction.

Infectious agents have had decisive influences on the history of mankind.%
Such is the
grand context for this modest book that concentrates on the language of
mathematical models and the tools for their analysis. We hope it will help
our readers to probe the intricacies of the real world.

It is a healthy attitude to compare models with data. But, we claim, insight
also very often derives from comparing models to models. We see the danger
though. In 1965, D.G. Kendall wrote, referring in particular to A.G. McKendrick and
R. Ross, who shaped epidemic theory in the early twentieth century using
their medical background as a starting point:

\begin{quote}
`Mathematicians may be blamed for subsequently carrying the game too
far, but its highly respectable medical origin should not be overlooked.'
\end{quote}

\noindent We wonder whether this cap of carrying the game too far fits us.
We do not think so and certainly do not hope so, but it is you, reader, who
decides.

Biological reality is complex and mathematical models are only caricatures
of it. Apparently Picasso once said:

\begin{quote}
`Art is a lie that helps us to discover the truth'
\end{quote}

\noindent In our context, `art' stands for simple models that describe
relations between key components of an essentially much more complex
reality. 

Finally, we give the floor to the medical doctor who arguably is the
founding father of modern epidemic theory, Sir Ronald Ross, who wrote (Ross, 1911, p. 651):

\begin{quote}
`As a matter of fact all epidemiology, concerned as it is with variation of disease from time to time
or from place to place, {\it must} (sic) be considered mathematically (...), if it is to be considered
scientifically at all. (...) And the mathematical method of treatment is really nothing but the
application of careful reasoning to the problems at hand.'
\end{quote}