Disease, an Integral Part of Differential Equations
MAT275 Honors Contract Project
By: Jared Duensing
Introduction:
The spread of diseases can often be modeled by scientists using mathematical representations, namely
differential equations. The best way to model such a scenario is by understanding that the rate at which
the disease spreads not only depends on the number of people that have already contracted the
disease, but additionally the people that have not already been affected. Obviously, what is at stake is
the fact that certain measures should be taken in certain circumstances. If a certain population contracts
a disease, it might be useful to model the information that is already known, that way the area can
determine whether or not extreme measures should be taken.
How it is done:
Various models can be used for various situations. For instance, in a closed environment, the spread of
disease can easily be modeled using the following equation:
I'(t)=r I(t) - a I(t)
t: time measured in days
I(t): number of people infected
r: rate of people becoming infected
a: rate of removal (this might be from recovery or death)
It can be noted from this equation that if the removal rate (a) is less than the rate of infected people (r),
I’(t), or presumably how the number of infected individuals changes, will be positive, therefore
indicating that the disease is spreading. If the opposite were to be the case, then the situation is under
control.
To find the general solution, I(t), the differential equation can be solved the following way:
dI/dt=(r-a)I(t)separable equation
dI/I=(r-a)dt integrate both sides
ln(I)=(r-a)t+cconstant results from integration without initial conditions
I=e^((r-a)t+c)to find particular solution, the natural log will be eliminated by using the exp. function
I=e^c*e^((r-a)t)
I(t)=Ce^((r-a)t)solution
Not surprisingly, the solution obtained for I(t) is the model traditionally associated with population
trends. Presumably, the constant obtained from integration is the initial amount of people infected,
while the exponential is being raised to the difference of the two rates at a certain instant of time. As
predicted, if r<a, the amount of people being infected is decreasing since the rate of infection is less
than the rate of recovery.
Several factors affect the two rates discussed in this report. For instance, with regards to r, this value
can be affected by factors such as the type of disease (how easily contagious it is) as well as contact
made by those who carry the disease. The rate “a,” on the other hand, can be affected by how fast
people are diagnosed and isolated from those that are healthy, for instance.
As an example, let’s say that in Ms. Brewer’s differential equations class, a student comes in
with a cold. Because the class is captivated by the lesson, the class lasts for about three days to
get a grasp on 3x3 systems of differential equations. (Now for this particular cold virus, it was
found following investigation in a different class that the rate, r, is 1/3 day^-1 and the rate, a, is
2/11.) The number of students infected is as follows:
I(3)=(1)e^((1/3-2/11)(3))=1.57 (or 2 students infected, maybe)
To look at a general graph of the scenario, the following is a graph of I(t)=(1)*e^((1/3-2/11)*t)
Essentially, if all thirty students were present, it would take roughly 22.5 days for a class of 30 to get
sick. Alternatively, if there were a cold virus going around and more than one person came in sick, say
three, that graph would look like the following:
And in this situation, it would only take roughly 15 days for the entire class to become sick. (This
scenario is obviously highly unrealistic. To better the results, the difference between r and a should be
increased so that the sickness is spread more rapidly to others).
Tweaking the values for r and a so that r<a presents a scenario in which the situation is being contained
and the disease or sickness is finally under control. If a certain city had 13,500 inhabitants sick, while the
rate of infection is ¼ day^-1 and rate of elimination is 2/5 day^-1, the situation can be modeled by the
following:
In this instance, it would take roughly 60-70 days for the last person to have contracted the disease to
either be cured (or perhaps die).
To conclude, dI/dt being positive indicates an epidemic. According to the models shown, this is impacted
by the values r and a--the rates that determine if a sickness is being contained, or if the situation is dire.
The problem with this model is that it is easy to test various values for “r” and “a” and notice the trends,
but it would be helpful to see from where these values arise. Fred Brauer and Carlos Castillo-Chavez
better help to explain these situations, using the example of a sexually transmitted disease. The number
infected again can be modeled by I(t), while they now define the transmission rate as β, the number of
people in a population is N, the rate of sexually active individuals that leave the sample area as µ and
the rate of those that join the sample area as γ. This now translates the equation I'(t)=r I(t) - a I(t) into:
dI/dt= β(N-I(t))*I/N-( µ+ γ)I(t)
or
dI/dt= β(1-I/N)*I(t)-( µ+ γ)I(t)
*Note that in this new equation, the “a” and “r” rates discussed before is broken up into rates that have
more components, making it more specific as to what is being included in the equation. In other words,
r can be equated to β(1-I/N), showing that the rate of infection can be split up into the percent of people
infected, I, over the total population size, N, multiplied by the transmission rate β. Similarly, the “a,” or
elimination rate, discussed earlier can be thought of as having two components, the rate of people
entering the infected zone, µ , plus the rate of people leaving the infected zone, γ, (that are not
infected.) By doing some substitutions, Brauer and Castillo-Chavez arrive at the following logistic
equation:
dI/dt=rI(1-I/K)=rI(t)-(I/K)*I(t) (which can be thought of as the initial equation explored, but more refined
and descriptive. Solving this differential equation will give a similar result: I(t)=Ce^(r-I/K)t
Conclusion:
Overall, many factors are involved in modeling sickness and disease. While bacterial growth is very
predictable, as there are very few factors that contribute to its reproduction and spread, the behavior of
humans is much more complex during times of disease that its spread can be influenced by any number
of factors. However, once accounted for, the rates involved in modeling this scenario make for a very
elementary differential equation showing the change in the number of infected individuals. By solving
this differential equation, one arrives at the all-too-familiar exponential function for modeling
population trends. The most striking aspect of this model is the rate involved, which consists of the
difference in the rate of infectious spread and the rate of eliminating the infection. What should be
noted is that if this value is negative, the disease is under control, but if it is positive, the sickness is an
epidemic.
Works Cited
1. Brauer, Fred, and Carlos Castillo-Chávez. Mathematical Models in Population Biology and
Epidemiology. New York: Springer, 2001. Print.
2. Burke, M. "Epidemiology Model (Differential Equations) Part I." College of Science &
Mathematics. Kennesaw University. Web. 25 Apr. 2011.
<http://science.kennesaw.edu/~mburke/modules/epidiffeq1.html>.
3. "Graphing Calculator." Holt McDougal Online. Web. 25 Apr. 2011.
<http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html>.