7
Basic Ideas of Mathematical
Epidemiology
7.1 Introduction
The idea of invisible living creatures as agents of disease goes back at least
to the writings of Aristotle (384 BC-322 BC). It developed as a theory in
the 16th century. The existence of microorganisms was demonstrated by
Leeuwenhoek (1632-1723) with the aid of the first microscopes. The first
expression of the germ theory of disease by Jacob Henle (1809-1885) carne
in 1840 and was developed by Robert Koch (1843-1910), Joseph Lister
1827-1912), and Louis Pasteur (1827-1875) in the latter part of the 19th
century and the early part of the 20th century.
The mechanism of transmission of infections is now known for most dis-
eases. Generally, diseases transmitted by viral agents, such as infl.uenza,
measles, rubella (German measles), and chicken pox, confer immunity against
reinfection, while diseases transmitted by bacteria, such as tuberculosis,
meningitis, and gonorrhea, confer no immunity against reinfection. Other
diseases, such as malaria, are transmitted not directly from human to hu-
man but by vectors, agents (usually insects) who are infected by humans
and who then transmit the disease to humans.
Communicable diseases such as measles, infl.uenza, or tuberculosis, are a
fact of modern life. We will be concerned both with epidemics which are
udden outbreaks of a disease, and endemic situations, in which a disease
is always present. The AIDS epidemie and outbursts of diseases such as the
Ebola virus are events of concern and interest to many people. The preva-
lence and effects of many diseases in less developed countries are probably
276 7. Basic Ideas of Mathematical Epidemiology
less well-known but may be of even more importance. Every year millions
of people die of measles, respiratory infections, diarrhea and other diseases
that are easily treated and not considered dangerous in the Western world.
Diseases such as malaria, typhus, cholera, schistosomiasis, and sleeping sick-
ness are endemic in many parts of the world. The effects of high disease
mortality on mean life span and of disease debilitation and martality on
the economy in affiicted countries are considerable.
Epidemiological models focus on the transmission dynamics of a trait or
traits transmitted from individual to individual, from population to pop-
ulation, from community to community, from region to region, or from
country to country. A "trait" can be a disease, such as (i) measles, HIV,
malaria, tuberculosis, (ii) a genetic characteristic, such as gender, race, ge-
netic diseases, (iii) a cultural "characteristic", such as language or religion,
(iv) an addictive activity, such as drug use, ar (v) the gain ar loss of in-
formation that are communicated through gossip, rumors, and so on. The
term "individual" in epidemiology is broadly understood to include various
epidemiological units. The selection of an epidemiological unit is based on
the question and the level of aggregation at which the investigator wishes
to address such a question. In the study of disease dynamics in the immune
system, types of cells provide the epidemiological unit; in the study of the
spread of malaria the host (humans ar other mammals) and the vectors
(female mosquitoes) are usually selected as the epidemiological units; in
the study of Chagas disease a "house" (infested houses may correspond ta
"infected" individuals) may be chosen as an epidemiological unit; in tuber-
culosis, a household ar community or a group of strongly linked individual
("cluster") may be the chosen unit.
The above are examples of epidemiological transmission processes which.
to be understood, must be studied from various perspectives that include
the study of their transmission dynamics at different spatial, temporal, or
organizational scales, the level of which is dictated by the question the
investigator wishes to address. Unfortunately the selection of a particular
level of organization and a model may determine a priori what is and wha
is not relevant. Hence, two relevant questions associated with epidemiolog-
ical processes are: (i) how much organizational detail such as population
structure, immune response, or genetic variability must be included in epi-
demiological models? and (ii) how is relevant detail modeled?
In this chapter we provide an introduction to mathematical epidemiolo -
which includes the development of mathematical models for the spread o:
disease as well as for their analysis.
An epidemie, which acts on a short temporal scale, may be describec
as a sudden outbreak of a disease that infects a substantial partion of tb
population in a region before it disappears. Epidemics usually leave marr:
members untouched. Often these attacks recur with intervals of severa
years between outbreaks, possibly diminishing in severity as populatiox-
7.1 Introduction 277
-elop some immunity. This is an important aspect of the connection
/een epidemics and disease evolution.
The Book of Exodus describes the plagues that Moses brought down
cpon Egypt, and there are several other biblical descriptions of epidemie
ubreaks. Descriptions of epidemics in ancient and medieval times fre-
• ently used the term "plague" because of a general belief that epidemics
represented divine retribution for sinful living. More recently some have
cribed AIDS as punishment for sinful activities. Such views have often
.zampered 01' delayed attempts to control this modern epidemie.
The historian W.H. McNeill argues, especialIy in his book Plagues and
Peoples (1976), that the spread of communicable diseases frequently has
been an important influence in history. For example, there was a sharp pop-
ation increase throughout the warld in the 18th century; the population
China increased from 150 million in 1760 to 313 million in 1794 and the
population of Europe increased from 118 million in 1700 to 187 million in
1800. There were many factors involved in this increase, including changes
in marriage age and technological improvements leading to increased food
supplies, but these factors are not sufficient to explain the increase. Demo-
graphic studies indicate that a satisfactory explanation requires recognition
of a decrease in the mortality caused by periodic epidemie infections. This
decrease carne about partly through improvements in medicine, but a more
important influence was probably the fact that more people developed im-
munities against infection as increased travel intensified the circulation and
co-circulation of diseases.
There are many biblical references to diseases as historical influences,
uch as the decision of Sennacherib, the king of Assyria, to abandon his
attempt to capture Jerusalem about 700 BC because of the ilIness of his sol-
diers (Isai ah 37,36-38). The fall of empires has been attributed directly ar
indirectly to epidemie diseases. In the 2nd century AD the so-called Anto-
nine plagues (possibly measles and smallpox) invaded the Roman Empire,
eausing drastie population reductions and economic hardships leading to
disintegration of the empire because of disorganization, which facilitated
invasions of barbarians. The Han empire in China collapsed in the 3rd cen-
tury AD after a very similar sequence of events. The defeat of a population
of millions of Aztecs by Cortez and his 600 folIowers ean be explained, in
part, by a smalIpox epidemie that devastated the Aztecs but had almost
no effect on the invading Spaniards thanks to their built-in immunities.
The Aztecs were not only weakened by disease but also confounded by
what they interpreted as a divine force favoring the invaders. Smallpox
then spread southward to the Incas in Peru and was an important factor in
the success of Pizarro's invasion a few years later. Srnallpox was folIowed
by other diseases such as measles and diphtheria imported from Europe to
Narth America. In some regions, the indigenous populations were reduced
to one tenth of their previous levels by these diseases: Between 1519 and
278 7. Basic Ideas of Mathematical Epidemiology
1530 the Indian population of Mexico was reduced from 30 million to 3
million.
The Black Death (bubonic plague) spread from Asia throughout Europe
in several waves during the 14th century, beginning in 1346, and is esti-
mated to have caused the death of as much as one-third of the population
of Europe between 1346 and 1350. The disease recurred regularly in vari-
ous parts of Europe for more than 300 years, notably as the Creat Plague
of London of 1665-1666. It then gradually withdrew from Europe. As the
plague struck some regions harshly while avoiding others, it had a profound
effect on political and economic developments in medieval times. In the last
bubonic plague epidemie in France (1720-1722), half the population of Mar-
seilles, 60 percent of the population in nearby Toulon, 44 per cent of the
population of Arles and 30 percent of the population of Aix and Avignon
died, but the epidemie did not spread beyond Provence. Expansions and
interpretations of these historical remarks may be found in McNeill (1976),
which was our primary source on the history of the spread and effects of
diseases.
The above examples depict the sudden dramatic impact that disease has
had on the demography of human populations via disease-induced mortal-
ity. In considering the combined role of diseases, war, and natural disasters
on martality rates, one may conclude that historically humans who are
more likely to survive and reproduce have either a good immune system, a
\;)I(ypellsityta avaid waI alld disasters, OI, nowadays, excellent medical care
and/ ar health insurance.
There are many questions of interest to public health physicians con-
fronted with a possible epidemie. Far example, how severe will an epidemi
be? This question may be interpreted in a variety ofways. For example, hor-
many individuals will be affected all together and thus require treatmen
What is the maximum number of people needing care at any particula:
time? How long will the epidemie last? How much good would quarantin
of victims do in reducing the severity of the epidemie?
For diseases that are endemic in some region public health physicians
need to be able to estimate the number of infectives at a given time as
well as the rate at which new infections arise. The effects of quaranti -
or vaccine in reducing the number of victims are of importance, just "-
in the treatment of epidemics. In addition, the possibility of defeating t "-
endemic nature of the disease and thus controlling or limiting the diseas
in a population is worthy of study. How can such questions be answered
Scientific experiments usually are designed to obtain information an
to test hypotheses. Experiments in epidemiology with controls are oft
difficult or impossible to design and even if it is possible to arrange an
periment there are serious ethical questions involved in withholding tr --
ment from a control group. Sometimes data may be obtained after the fac
from reports of epidemics or of endemic disease levels, but the data mz
be incomplete or inaccurate. In addition, data may contain enough irrezt-
7.1 Introduction 279
ies to raise serious questions of interpretation, such as whether there is
.~ence of chaotic behavior [Ellner, GaUant, and Theiler (1995)1. Hence,
rameter estimation and model fitting are very difficult. These issues raise
question of whether mathematical modeling in epidemiology is of value .
. .Iathematical modeling in epidemiology provides understanding of the
_~erlying mechanisms that influence the spread of disease and, in the
--cess, it suggests control strategies. In fact, models often identify be-
sviors that are unclear in experimental data-often because data are non-
ro duci ble and the number of data points is limited and subject to errors
_ measurement. For example, one of the fundamental results in mathemat-
epidemiology is that most mathematical epidemie models, including
e that include a high degree of heterogeneity, usuallyexhibit "thresh-
d" behavior which in epidemiological terms can be stated as follows: lf
-~e average number of seeondary infeetions eaused by an average infeetive
> less than one a disease will die out, while if it exeeeds one there will be
n epidemie. This broad principle, consistent with observations and quan-
zified via epidemiological models, has been consistently used to estimate
the effectiveness of vaccination policies and the likelihood that a disease
ay be eliminated or eradicated. Hence, even if it is not possible to verify
.ypotheses accurately, agreement with hypotheses of a qualitative nature
" often valuable. Expressions for the basic reproductive number for HIV
in various populations is being used to test the possible effectiveness of
vaccines that may provide temporary protection by reducing either HIV-
infectiousness or suceptibility to HIV. Models are used to. estim.ate -how
widespread a vaccination plan must be to prevent or reduce the spread of
HIV.
In the mathematical modeling of disease transmission, as in most other
areas of mathematical modeling, there is always a trade-off between simple
models, which omit most details and are designed only to highlight gen-
eral qualitative behavior, and detailed models, usually designed for specific
ituations including short-term quantitative predictions. Detailed models
are generally difficult or impossible to solve analytically and hence their
usefulness for theoretical purposes is limited, although their strategic value
may be high. This chapter begins with simple models in order to establish
broad principles. Furthermore, these simple models have additional value
as they are the building blocks of models that include detailed structure.
A specific goal of this chapter is to compare the dynamics of simple and
slightly more detailed models primarily to see whether slightly different
assumptions can lead to significant differences in qualitative behavior.
We will often think of a disease as an invasion of a host population,
consisting of separate patches (individuals), by a pathogen. An epidemie
is a successful invasion if the number of occupied patches increases over
time after the initial introduction of the pathogen into the host (patch)
population.
280 7. Basic Ideas of Iathernatical Epidemiology
dany of the early developments in the mathematical modeling of corn-
municable diseases are due to public health physicians. The first known
result in mathematical epidemiology is a defense of the practice of inocula-
tion against smallpox in 1760 by Daniel Bernouilli, a member of a famous
family of mathematicians (eight spread over three generations) who had
been trained as a physician. The first contributions to modern mathemat-
ical epidemiology are due to P.D. En'ko between 1873 and 1894 [Dietz
(1988)]), and the foundations of the entire approach to epidemiology based
on compartmental models were laid by public health physicians such as Sir
R.A. Ross, "V.H. Hamer, A.G. McKendrick, and W.O. Kermack between
1900 and 1935, along with important contributions from a statistical per-
spective by J. Brownlee. A particularly instructive example is the work of
Ross on malaria. Dr. Ross was awarded the second Nobel Prize in Medicine
for his demonstration of the dynamics of the transmission of malaria be-
tween mosquitoes and humans. Although his work received immediate ac-
ceptance in the medical community, his conclusion that malaria could be
controlled by controlling mosquitoes was dismissed on the grounds that it
would be impossible to rid a region of mosquitoes completely and that in
any case mosquitoes would soon reinvade the region. After Ross forrnu-
lated a mathematical model that predicted that mala.ria outbreaks could
be avoided if the mosquito population could be reduced below a critica!
threshold level, field trials supported his conclusions and led to sometimes
brilliant successes in malaria control. However, the Garki project provides a
dramatic counterexample. This project worked to eradicate malaria from a
region temporarily. However, people who have recovered from an attack of
malaria have a temporary immunity against reinfection. Thus elimination
of malaria from a region leaves the inhabitants of this region without im-
munity when the campaign ends, and the result can be a serious outbreak
of malaria.
We formulate our descriptions as compartmental models, with the pop-
ulation under study being divided into compartments and with assump-
tions about the nature and time rate of transfer from one compartmen
to another. Diseases that confer immunity have a different compartmental
structure from diseases without immunity and from diseases transmitted by
vectors. The rates of transfer between compartments are expressed math-
ernatically as derivatives with respect to time of the sizes of the compar -
ments, and as a result our models are formulated initially as differenti
equations. Later, when we study models in which the rates of transfer de-
pend on the sizes of compartments over the past as well as at the instan;
of transfer, more general types of functional equation, such as differentia.-
difference equations or integral equations, will be used.
