Case Studies and Project Ideas: Small Pox

Disease Background:

Smallpox is a highly contagious disease which is often fatal in humans. It is well recognized by the distinctive rash that develops on the skin of patients which often results in permanent deep-pitted scars. Patients also usually run extremely high fevers while infected with the disease. Smallpox, because of its highly contagious nature, has been the cause of epidemics around the world. Since it is caused by a certain virus, smallpox can be contracted by simple contact with infected persons or contaminated objects. Once infected, hardly more than 25% of patients live. However, if a patient does survive, that person's body develops an immunity to any further infection by the virus.

Several ways have been developed through the years to combat the smallpox disease. Early in the eighteenth century, this was done through inoculation. Inoculation involves applying a piece of skin from an infected person to the skin of a healthy person. By doing this, the healthy person would develop of mild case of smallpox that would create an immunity to the virus. This is very dangerous though, since the healthy person is exposed to the actual virus and this alone could cause a severe and perhaps fatal case of smallpox.

Today, smallpox is combated through vaccination. In 1796, the English medical student Edward Jenner found that an infection of the relatively mild cowpox virus also resulted in an immunity to smallpox. This was much safer than exposing humans to the actual smallpox virus. Modern vaccinations are now extremely safe and involve injecting a patient with simply a mild strain of the cowpox virus.

Epidemiology

Smallpox has long been an interesting topic for mathematicians to model, primarily because of the huge epidemics that occurred in Europe during the eighteenth century. There was significant disagreement in these earlier times about whether the risks of death from inoculation were worth the benefits of immunity from smallpox. Daniel Bernoulli attempted to study this question by using a mathematical model.

Bernoulli's model was very simple. It involved only a healthy population, an infected population, and an immune population. Based on observed evidence, Bernoulli estimated the rates at which people became infected and the chances that these infected people could possibly become well on their own.

He then considered the effects of inoculation. What would happen if the healthy population were allowed to become immune to the smallpox virus? This would have to consider the fact that inoculating people could result in possibly killing them. By using his model, Bernoulli calculated that the fatality rate of inoculation must be no more than 1 out of 200 before the technique should be used. Often of course, the rate was much higher and Bernoulli's work sparked controversy.

Today, vaccination replaces inoculation. This technique is very safe and immunization is virtually assured. The main problem here is cost. How widespread and under what circumstances should vaccinations be used for the resulting benefits? In the USA, for example, there was a huge vaccination drive in 1968. 14,168,000 people were vaccinated at a cost of $150,000,000. 572 people suffered complications and 9 people died. However, not a single case of smallpox was reported from 1950-1970. Many people, though, had to suffer needlessly and the millions of dollars spent could have been used elsewhere for such things as education. Thus, it is probable that vaccinations on this scale may not be worth the cost in a developed country like the USA.

Sudan: A Case Study

Sudan, a country in eastern Africa, presents an interesting study of smallpox epidemics. It is a developing country located primarily between Egypt, Chad, Ethiopia and the Red Sea. The capital of the nation is Khartoum with a population of around half a million. In the 1950s, Sudan overall had approximately 8 million people, but the country now has over 20 million. This can be attributed largely to the fact that during the 1950s, Sudan had a extremely high yearly birth rate of about 50 births per 1000 people as well as a large refugee population entering from other countries. The population has exploded despite Sudan having quite a high yearly death rate of 20 deaths per 1000 people. Much of the country is remote and medical facilities are in scant supply. Thus dealing with epidemics such as smallpox is a very difficult challenge to the leaders of this country. Diseases in the area typically sweep from east to west, from Chad to the Red Sea, along a corridor known as the "epidemic belt" of the Sudan.

One the most recent major modern epidemics occurred in the years 1953-1956. In 1951 to 1954, smallpox made a comeback in Sudan--coinciding with an epidemic in neighboring Chad. In the early months of 1952, 72 cases were found among immigrants in the western tip of the country. Soon the disease spread east peaking finally in 1952. 3,653 cases resulting in 578 deaths from smallpox were reported that year. Interestingly enough, the disease was partially stopped from spreading further east by a well placed immunized zone across the center of the country.

Reported Smallpox Cases in Sudan 1950-1960

Year	Number of Cases
1950-1951	181
1951-1952	346
1952-1953	3,670
1953-1954	3,030
1954-1955	4,200
1955-1956	1,427
1956-1957	25
1957-1958	295
1958-1959	380
1959-1960	336

This chart shows how the number of cases of smallpox climbed and then dropped during the 1953-1956 epidemic. During the 1950s, annual vaccinations in the country were increased to 1,259,063 per year in an effort to curb the epidemic. As seen from the later years of the decade, this effort was effective. However, smallpox has a record of flaring up periodically throughout Sudan's history and could do so again.

Analysis:

With STELLA it is possible to make Bernoulli's model even more complex. Try to apply Bernoulli's primarily European model to the case study in Sudan. What happens when new births and deaths from natural causes are introduced into the model? At what point does the risk of death from inoculation become acceptable? Change the model so that vaccination replaces inoculation. Allow the model to calculate the costs of the resulting vaccinations and see how expensive this becomes.

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