A Rumor Can Be Epidemic!

Joseph DiCarlucci

Abstract

This project will describe several mathematical models for the background information and pedagogical ideas that can be molded into exploratory lessons based on actual modeling techniques for epidemics.

Modeling a Rumor

Suppose there is a town with population ( ) where a rumor spreads by word of mouth. Each inhabitant who has heard the rumor continues to tell it until he meets someone who has already heard it. Then he stops telling it. Now, divide the town's population into three distinct groups:

= all those who have not heard the rumor (susceptible)

= all those who are spreading the rumor (infectious)

= all those who no longer spread the rumor (removed or immune)

Hence, in mathematical symbols,

Assume that every pair of individuals has equal probability of coming into contact with each other. In other words, the three groups are uniformly mixed. The rumor spreads through SI-encounters where I tells S the rumor then S joins I in the infectious group that is spreading the rumor. The number of possible SI-encounters is . The change in S is proportional to . If the infection rate of the rumor is , then the change in group S can be described by the difference equation in the time interval . Individuals become immune to the rumor (stop spreading it) in one of two ways, either when two persons spreading the rumor meet each other (II- encounter) or when a person spreading the rumor meets someone who has stopped telling it (IR-encounter). There are possible IR-encounters and possible pairs of II-encounters. Since II-encounters remove two people from actively spreading the rumor, then the number of removals is proportional to . Remember, however, that group I is also simultaneously increasing its number by SI-encounters. Let us assume the removal rate is equal to the infection rate (). So, in time interval , . This last equation can be simplified algebraically, since using . Consequently, . There is no reason to develop a corresponding difference equation for, since the population of the town is constant .

Model 1.

The dynamics of the rumor can be visually illustrated through the TI-82 calculator in the SEQ mode. Let the initial conditions be . The two sequence functions for , respectively, for Model 1 are:

Store the initial values for P and A, , then set U_start=99 and V_start=1. The rumor ends when (approximately 12 iterations). The graphs are displayed in the figure below, and the table of values can be searched for numerical analysis.

Questions:

1) Why do we stop the model when ?

2) About what percent of the population never hears the rumor?

3) Find the maximum number of infectives, I_max, during the rumor epidemic? On what day did the I_max occur?

4) About how many immune individuals were there when the rumor ended?

5) If we let the model continue to run, what values will approach? Hint: increase number of iterations.

6) The severity () of the rumor epidemic can be calculated by . Find F for the model above, then explain why this definition of severity is reasonable.

7) How can we "fine-tune" the model to improve our data values?

The answer to question #7, admittedly, depends on the computational tools available and the planned use(s) for the model. Yet, we can "fine-tune" the model by decreasing from 1 to, say, 0.5. In general, the model's functions are given by:

Exercise 1

Construct the sequence functions on the TI-82 calculator for Model 1 using the same given values, then generate their corresponding graphs and tables when . How many iterations are needed in each case so that ? Answer question #3 again for and. In which result do you have the most confidence? Why?

Since the number of required iterations is not transparent when setting the SEQ WINDOW, the following TI-82 program will allow entry of initial values and output the necessary number of iterations and .

PROGRAM:RUMORS

:0K

:DISP "INPUT A":INPUT A

:DISP "INPUT P":INPUT P

:DISP "INPUT DELTA T":INPUT T

:DISP "INPUT I(0)": INPUT I

:P-IS

:LBL 1

:S - ASITW

:I - AIT(P - 2S - 1)I

:WS:1 + KK

:IF I 1: THEN:GOTO 1:END

:P - (S + I)R

:DISP "S(N)",S

:DISP "I(N)",I

:DISP "R(N)",R

:DISP "NO. OF ITERATIONS", K

Exercise 2

Experiment, using the TI-82 sequence functions and the program RUMORS, by changing the parameter values . Examine the graphs and table values for each incremental change to a single variable. Can you find values for the initial conditions so that a stable condition exists (relatively no change in S or I)? Can you predict if a rumor will spread or not? Can you find initial values that cause the rumor to become endemic?

Whether or not a rumor will spread is a critical concern. A rumor will spread as long as the infectious group (I) is growing by gaining new members. Mathematically, this can be represented as

Without loss of generality, let , then

We want to know when

which occurs when

This value,

is called the rumor threshold. Consequently, if

then rumor spreads (I increases), at

then rumor peaks (maximum I), and if

then rumor declines (I decreases). Moreover, if is less than and =1, no epidemic ensues and is a stable point. For example, why is it that when an epidemic goes through a school, there are some who survive as susceptibles? The calculation of the threshold value shows how this can happen. If is near or below the threshold, then few or no susceptibles will become infected because there is not enough contact or the infectives are removed quickly enough. In short, the critical density of susceptibles necessary to maintain a rumor in Model 1 is

Extension 1. In Model 1, the infection rate = removal rate = .

Suppose the rates are different; let = infection rate and = removal rate.

Construct a new model (2) using and , and find the rumor threshold for .

Model 2.

Exercise 3

In the March 4, 1978 issue of the British Medical Journal there was a report with detailed statistics of a flu epidemic in a boys' boarding school with a total of 763 students. If , find the epidemic threshold, the maximum number of infectives, the total number of unaffected boys, and the severity of the epidemic.

Exercise 4

Modify program:RUMORS to fit Model 2, and include as an output the measure of severity (F).

Extension 2. There are rumors in which removals can become susceptible again. The model for a recurrent rumor is given below in Model 3, where infection rate and removal rate = recurrent rate. Many sexual diseases are recurrent in nature, such that a person who leaves the infectious class is immediately susceptible again.

Model 3.

In this model, . There are two interesting cases that need investigation. When the threshold value , then . Try several experiments to convince yourself of this fact; for instance, let . If the threshold value , then converges to the threshold value. When this condition exists, the rumor is not epidemic but, rather, endemic.

Bibliography

Engel, Arthur. Elementary Mathematics from an Algorithmic Standpoint. Staffordshire, UK: Keele Mathematical Education Publications, 1984.

Hoppensteadt, F. C. and C. S. Peskin. Mathematics in Medicine and Life Sciences. NY: Springer-Verlag, 1992.

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