138 Ordinary Differential Equations
3.4 Simple Epidemic Models
One of the greatest causes of human suffering and misery is the outbreak of
various types of epidemics. Over the ages, numerous deaths have been caused
by epidemics and frequently a large proportion of a community has perished.
Today well-developed countries are relatively free from the threat of death
producing epidemics; however, epidemics of influenza still occur occasionally.
Less well-developed countries in Africa and the Far East are still susceptible
to lethal epidemics.
One of the earliest recorded accounts of an epidemic occurred in Athens
from 430 B.C. to 428 B.C. The "Golden Age" of Athens coincided closely
with the reign of Pericles, who rose to power in 469 B.C. The Peloponnesian
War began in 431 B.C. and lasted nearly one quarter of a century. The
war was essentially a conflict between the Greek city-states of Athens and
Sparta. Athens had a strong navy and a weak army, while Sparta had a strong
army and a weak navy. Pericles decided to bring the people from the a reas
surrounding Athens into the fortified city to protect his state from attack by
land. Simultaneously, he had his navy attack the coastal areas of the Spartan
state. This strategy worked well during the first year and it appeared Athens
would soon win the war. However, overcrowding and unsanitary conditions
produced an outbreak of a highly contagious disease. Victims of the disease
usually died within six to eight days after being infected and all toll between
30% and 60% of the population of Athens perished due to the disease. When
the epidemic of 430 B.C. ended, Pericles sent his navy to capture the Spartan
stronghold of Potidaea. The dreaded disease struck the crews whil e they
were still at sea and forced them to return to Athens prior to accomplishing
their mission. The epidemic swept through Athens again in 429 B.C. and in
428 B.C. when Pericles died from it. The war, itself, continued for several
more years and finally the Spartans, aided by the epidemics, were able to
defeat the Athenians.
In the fourteenth century an epidemic of the bubonic plague killed ap-
proximately one-fourth of the population of Europe, which was estimated to
number 100 million. In 1520 a smallpox epidemic caused the death of half
of the Aztec population of 3.5 million. An epidemic of measles on the island
of Fiji in 1875 resulted in the death of 40,000 people out of a population of
150,000. From 1918 to 1921 there was a typhus epidemic in the Soviet Union
which killed approximately 2.5 million people. In 1919 in a worldwide in-
fluenza epidemic, an estimated 20 million people perished from the disease or
the complication of pneumonia.
In the seventeenth century John Graunt (1620-1674) and Sir William Petty
(1623-1687) collected information on the incidence and location of epidemics.
On April 30 , 1760, Daniel Bernoulli presented a paper to the Academie Royale