Module 2: Epidemics and the Spread of Diseases

Introduction

An epidemic is the spread of an infectious disease through a community affecting significant numbers of people in that community. Examples of epidemics are colds, flu, measles, cholera, etc. Among the most notorious of epidemics are: the "Black Death" (bubonic plague) in Asia and Europe in the 14th century; the 1918 influenza epidemic which killed tens of millions; and the poliomyelitis epidemic of the 20th century. Recent attention has focused on epidemics caused by the Ebola virus and the HIV virus. The study of epidemics is concerned with the transmission of a disease from an infected host to an uninfected susceptible person. The transmission may result from direct contact between individuals or it may involve an intermediate host such as a mosquito in the case of malaria attacks epidemics. Another mode of transmission might be a virus in the air or in the water. Normally in the spread of an epidemic, the number of infective persons tends to rise sharply at first and then tapers off as the disease runs its course or is brought under control. One complicating factor is the existence of "carriers" who are healthy individuals who transmit the disease to others. Further complications result from the fact that some diseases make the victims immune from further attack for life.

The following are typical questions relative to disease transmission which health officials must answer:

1) If an infection is introduced into a community at what rate will it spread?

2) How many initial infections are required to cause an eventual epidemic?

3) When will the epidemic be at its worst?

4) Does the number of infected people approach some maximum value?

5) Does the number of people infected follow any specific pattern?

The task of this module is to consider some of these questions using actual data and to construct mathematical models to describe the spread of an infectious disease through a community.

A Graphical Introduction

Consider the two graphs below which provide data on the annual number of reported cases for measles and whooping cough in a certain region.

a) What patterns can you detect concerning the incidence of the diseases?

b) What long term trends can you observe?

c) Discuss how such factors as living conditions or the introduction of vaccines

might affect the data.

d) Whooping cough vaccination was first recommended for children in this region

in 1957. Is this fact reflected in the data?

e) In 1974 a report was published which suggested that the whooping cough

vaccine might have dangerous side effects. Is this fact reflected in the data?

Creating a Model

In an attempt to develop a simple model for the spread of diseases we first assume that a population experiencing an epidemic can be separated into three groups. (See Part 2, A General Model for Epidemics, of Case Study 7, page 103 of Berry & Houston.)

S: Susceptibles, people who are currently uninfected but who may become infected

I: Infectives, people who are infected and capable of spreading the infection (infectious)

R: Removals, persons who died from the disease or who recovered and are now immune.

Let N represent the size of the population and n represent the number of days (or weeks) measuring from some starting day. Let:

x_n denote the number of persons in group S on day n,

y_n denote the number of persons in group I on day n,

z_n denote the number of persons in group R on day n.

Under the assumption that there are no births or deaths due to other causes during the epidemic, we have

N = x_n + y_n + z_n .

It is reasonable to assume that at the beginning of the epidemic, at n=0, that x₀ is large, that y₀ is fairly small, and that z₀ = 0. The following assumptions which are empirically based provide the basis for our model:

a) The percentage of susceptibles who contract the disease each day is proportional to the number of infected people. Symbolically:

x_n+1 - x_n = -ky_n k > 0

x_n

Thus we see that the change in the number of susceptibles, S, from day n to day n+1 is proportional to the product of the susceptibles and the infectives on day n:

x_n+1 - x_n = -kx_ny_n k > 0

b) A constant percentage of infected people are removed (die, recover, or become immune) from the diseased each day. Symbolically:

z_n+1 - z_n = r r > 0

y_n

This says that the change in the number of removals, R, from day n to day n+1 is proportional to the number of infectives on day n, i.e.,

z_n+1 - z_n = ry_n r > 0.

c) From a) and b) we deduce that the change in the number of infectives, I, from day n to day n+1 is given by:

y_n+1 - y_n = kx_ny_n - ry_n.

In summary, a discrete-time model for an epidemic under the above simplifying assumptions is given by the difference equations:

(1) x_n+1 = x_n - kx_ny_n Susceptibles

(2) y_n+1 = y_n + kx_ny_n - ry_n Infectives

(3) z_n+1 = z_n + ry_n Removals.

This model is particularly useful because if the constants of proportionality, k and r, are

known for a population, values of x_n, y_n, and z_n can be generated on a spread sheet from initial values.

Tutorial Example

Suppose that in a population of 1000 people 20 people are initially immune from a disease and one person is initially infected (at day zero). Generate values of x_n, y_n, and z_n if it is known that k = .0005 and r = .05. Display your results in a table of values for x_n, y_n and z_n, and generate graphs of each function. How many days will it take for the susceptibles to be eradicated? On what day did the epidemic produce the largest number of infectives? Approximately how long does the epidemic last?

A. Tutorial Problem : Spread of a Cold ( Case Study #7, P. 102, Berry & Houston)

About every six weeks a ship arrives at an isolated island. Sometimes one of the islanders catches a cold from a sailor and it spreads through the island. The data below come from one particular epidemic. Use the data to develop a graphical model of the spread of a cold through the island community and make a list of observations about the epidemic from your model. You might consider two different graphs: one of the actual data and one showing a running total of the accumulated cases. Use the data to estimate the parameters k and r in the model and generate values of the susceptibles, infectives and removals for this epidemic.

Spread of a Cold Data

Day	New Cases	Day	New Cases
1	1	11	3
2	1	12	4
3	1	13	2
4	0	14	2
5	8	15	1
6	8	16	0
7	15	17	0
8	4	18	1
9	23	19	0
10	5	20	0

B. Tutorial Problem: Measles Outbreak (#17 P. 179, Huntley and James)

The data in the table below are for a measles outbreak in 1980 for a large city with a population of about 300,000. The table gives the number of new cases c_n reported in each week n.

n	cn	n	cn	n	cn	n	cn
0	6	10	46	20	67	30	26
1	7	11	45	21	28	31	36
2	13	12	66	22	41	32	35
3	13	13	60	23	42	33	21
4	17	14	82	24	34	34	12
5	32	15	130	25	55	35	9
6	21	16	98	26	27	36	8
7	26	17	82	27	40	37	7
8	29	18	54	28	33	38	1
9	27	19	93	29	15	39	1

Make a plot of this data against n, the week number, and record your observations. Make appropriate assumptions on the size of the population at risk and estimate the parameters k and r for this epidemic. Generate values of x_n, y_n, and z_n on a spread sheet and sketch graphs of the latter.

C. Tutorial Problem: Plague of Bombay (Bartkovich, P. 458)

The table below gives the number of death per day for a plague similar to one that struck Bombay, India in 1905. Use this data to estimate the values of k and r for this disease. Then use the estimated values of k and r to generate by spread sheet values of x_n, y_n, and z_n. Sketch graphs of the latter values.

Day	Deaths	Day	Deaths	Day	Deaths
1	20	11	450	21	370
2	20	12	750	22	220
3	30	13	770	23	100
4	40	14	700	24	80
5	60	15	690	25	60
6	50	16	860	26	40
7	110	17	920	27	60
8	160	18	810	28	50
9	280	19	590	29	50
10	390	20	400	30	30

D. Applications

1. Ebola Outbreak of 1995 in Zire

The graph below gives the number of deaths each day attributed to the Ebola virus in a province in Zaire during a three month period in 1995. (Note this data was acquired from the Internet using the address indicated on the graph. Students should visit this address for additional details about the epidemic.) Use the graph to develop a table of deaths per day due to the virus. Then estimate values of the parameters k and r in our model and generate the appropriate graphs. (Use an estimate of N=900 for the number of people at risk in the province.)

2. The AIDS Epidemic

The following data were reported in the August 1991 issue of HIV/AIDS Surveillance published by the Centers for Disease Control (CDC) in Atlanta. The Cases column gives the number of AIDS cases diagnosed in the designated interval. In this application students are asked to predict the trend of future AIDS cases by fitting an exponential curve to the data and a cubic curve to the data. Perform error analysis to see which type of curve gives the better fit. Using your experimental curve, indicate the number of AIDS cases that your experimental curve predicts for December 1996. Then compare with the actual number of AIDS cases reported by the CDC.

Half-Year Cases Cumulative

___________________________________________________________________

1981 Jan-June 92 92

July-Dec 203 295

1982 Jan-June 390 685

July-Dec 689 1374

1983 Jan-June 1227 2651

July-Dec 1642 4293

1984 Jan-June 2550 6843

July-Dec 3368 10211

1985 Jan-June 4842 15053

July-Dec 6225 21278

1986 Jan-June 8215 29493

July-Dec 9860 39353

1987 Jan-June 12764 52117

July-Dec 14173 66290

1988 Jan-June 16113 82403

July-Dec 16507 98910

1989 Jan-June 18452 117362

July-Dec 18252 135614

1990 Jan-June 18601 154215

July-Dec 16236 170851

1991 Jan-June 12620 183471

Notes:

1. This table includes only adults and adolescents. There have been 3199 cases

reported among children less than 13 years old.

2. 85 cases were reported before 1981.

3. The table gives the number of AIDS cases diagnosed, not the number of deaths.

The August issue reports that 118411 individuals have died from AIDS.

4. The last two numbers in the Cases column are almost certainly too low, probably

Due to delayed reporting. They should be omitted in the curve-fitting.

3. Viral Loading Investigation

This section begins with a lecture by Dr. Mildred Fuller on the HIV virus.

1. Describe how the HIV (the AIDS virus) reproduces itself in an infected cell.

2. Describe the methods used to detect the amounts of the virus in the blood.

3. What is viral load and what is its purpose?

4.How does the polymerase chain reaction (PCR) quantify viral burden?

5. What relationship does CD4 count have with viral load.

6. What are protease inhibitors? Explain their role in reducing the AIDS virus in the

blood.

7. Describe briefly the result of the Multi-center AIDS Cohort Study.

8. Describe how viral load determinations are useful in predicting the success or

failure of specific treatments for AIDS.