Reed-Frost Epidemic Model

INTRODUCTION

Epidemics of infectious diseases have afflicted and perplexed humans throughout history. Infectious diseases, and the epidemics they cause, come and go, and humans cope with them in different ways, but the fear and confusion over new infectious diseases, such as AIDS, and even measles, is an ancient pattern we have seen many times before.

Centuries ago, human efforts to combat epidemics focused on avoidance (or elimination) of active cases. Next came treatment of active cases, and within the last two hundred years, scientific advances made possible prevention of certain diseases by means of vaccines. But it was not until this century that people started to study the epidemic process itself - understanding the pattern of how diseases spread through a population. This field of study is known as "epidemiology" and it is found mainly in medical schools and the Center for Disease Control in Atlanta.

A measles epidemic in an elementary school offers a good example of questions epidemiology ties to answer. Suppose a school has 500 students. Some of the students are immune to measles because they have already had measles or because they have been vaccinated. The remaining students are susceptible to measles and are referred to as "susceptibles." In this example, let's suppose there are 40 immunes and 400 susceptibles. If we introduce an active case of measles into this school, what patter, if any, will the epidemic follow? In other words, if the principal asks us how many students will be absent on the fourth day of the measles epidemic, what should we tell him/her? How many children will not contract measles during this epidemic

To help answer such questions, two medical researchers at John's Hopkins University, Lowell Reed and Wade Hampton Frost, developed a mathematical model to describe accurately how diseases spread through populations. Their model, developed in the 1920's, has come to be known as the "Reed-Frost Epidemic Model." Their purpose in developing this model was to sensitize medical students to the variability of the epidemic process. Neither Reed nor Frost through their model worthy of publication, so the model is described by another author (Abbey) as follows:

  • The infection is spread directly from infected individuals to others by a certain kind of contact ("adequate" or "effective" contact) and in no other way.
  • Any susceptible individual in the group, after such contact with an infectious person in a given period, will develop the infection and will be infectious to others only within the following time period, after which he (she) is wholly immune.
  • Each individual has a fixed probability of coming into adequate contact with any other specified individual in the group within one time interval, and this probability is the same for every member of the group.
  • The individuals are wholly segregated from others outside the group.
  • These conditions remain constant during the epidemic.

In other words, each individual in the study population is in one of three possible states during any time period. These are:

  1. active case state
  2. susceptible state
  3. immune state

Active cases always change to the immune state in subsequent time periods. Immune individuals remain immune and are sometimes omitted from consideration. Susceptible individuals change to active cases if and only if they come into "effective contact" with active cases.

The traditional notation for the model (described by Fine) is as follows:

  • St = Number of susceptibles at time t.
  • Ct = Number of active cases at time t.
  • p = Probability that any two individuals selected at random will come into effective contact (contact sufficient for an active case to infect a susceptible individual).

Using this notation, Fine observes that the probability of a susceptible individual contracting the disease during time period t is:

1-(1-p) Ct

By substituting q for 1 - p, the above can be written:

1-qCt

By applying the above probability to a binomial probability distribution (the same probability distribution that predicts the expected number of heads when flipping coins), the expected number of cases can be computed easily for any time period. But this mathematical method lacks intuitive appeal and fails to convince those untrained to probability and statistics. In addition, it makes the unrealistic assumption that every individual encounters every other individual in one time period. In particular, the mathematical solution fails to sensitive physicians (and patients alike) to the great variability to epidemic patterns that are attributable to chance alone.

In short, Reed and Frost developed a wonderful technique for studying and demonstrating epidemic patterns. They gave us the mathematical model which is accurate but hard to understand for laymen.

Our challenge to students in the Summer Institute is:

Use modern computer technology to revitalize the old, but very good ideas of Reed and Frost!

THE PROJECT

Problem 1

Write a program on a workstation that emulates the mathematical epidemic model of Reed and Frost. For purposes of simplicity, fix the number of individuals in the population at 1,000 and fix the maximum number of time periods at 25. However, allow the probability of effective contact and the number of immunes to vary s that the effects of social service programs such as immunization or prevention education can be observed. Thus, the only input to your program will be the initial number of immunes and the effective contact probability.

For output, your program should display in each time period the number of:

  1. susceptibles

  2. active cases

  3. immunes

Run the program many times while varying the number of immunes and the probability of effective contact. Sensitize yourself to epidemic patterns and variations.

Problem 2

Write a program on the workstation that simulates a modified form of the Reed-Frost epidemic model. For this modification, assume that all the individuals are standing next to each other in a room.

Like before, some will be immune. Let one individual be an active case. He can now infect only his immediate neighbors. In the next time period, those people who got sick now have a chance to infect their neighbors and so on.

You will need the same input and output that was described in Problem 1. In addition you may want to input the position of that first active case.

The most dramatic output, though, will be a two-dimensional grid that shows all the individuals in the room and what state they are in (i.e., sick, immune, or contagious). As time periods go by, you can watch the disease spread through the room.

Run your program many times with different input parameters. How does the pattern of the epidemic flow change with affective contact probability?

Problem 3

Write a program on the CRAY based on the one you wrote for problem 2 except add graphic output using AVS that is a significant improvement over the numeric output from the workstation. You have seen one approach in the video; other approaches exist. Be ingenious! For example, beyond the tree states specified by Reed and Frost, the video introduced a fourth group that Reed and Frost neglected; those infected by the current epidemic. This is one form of enhancement you might consider.

Another enhancement that could prove interesting is to allow some individuals in the room to move. What do you expect to happen if a contagious person happens to move to another part of the room? How will the speed and spread of the disease change?

Good luck and have fun!

ANIMATIONS

SI 1998

REFERENCES

Abbey, H,: An examination of the Reed Frost theory of epidemics. Human Biology, 24:201-233, 1952.

Fine, Paul E.M.: A commentary on the mechanical analogue to the Reed-Frost epidemic Model. American Journal of Epidemiology, V106, Ns, 87-100, 1977.

Dave Ennis is the OSC coordinator for the Epidemic project. Dave's office is in 420-4. Please contact Dave to set up appointment(s) for consultation.

For assistance, write si-contact@osc.edu or call 614-292-0890.