The Possum Plague: Disease Dynamics 
-----------------------------------
            

x' = c*x - d*y // total Possums 
y' = y*(x - y) -r*y // Infected Possums


/* These equations model the dynamics
of the spread of tuberculosis in the 
New Zealand possum population. The 
possum was introduced into New Zealand
in 1830 to stimulate the fur trade.
There were no natural predators of the
possum on the island, and they soon
proliferated. There are now more than 
70 million possums in New Zealand! 
Unfortunately, possums can carry a 
form of tuberculosis (about half of
them are infected) that can be
communicated to livestock. Since 
cattle ranching is a very important
part of the New Zealand's economy,
there is a major effort underway to
understand the ecology of the possum
and the dynamics of the disease. The
simplest of these models is treated
here.

The equations given above are scaled
versions of the original model which
directly incorporates the biology.
Those equations are:

P' = (a - b)P - uI
I' = vI(P - I) - (u + b)I
// P = total possum population
// I = infected possum population

where a and b are the natural intrin-
sic birth and death rates of possums,
u is the disease-induced death rate
constant, and v is the infectivity 
which measures the efficiency with
which the population of infected
possums (size I) communicates the
disease to the population of healthy
possums (size P - I). Once a possum
catches the disease, it does not
recover.  Thus P is the size of the
total possum population and P >= I. 

For our purposes, think of x as a
scaled version of P and y as a scaled
version of I. The parameters c,d and r
are, respectively, scaled versions of

1) c=(a-b)/v -intrinsic natural growth 
2) d=u/v - disease induced attrition
3) r=(u+b)/v - combined attrition

For more information about this model,
read the article by G. Wake in the 
Winter 1995 issue of the CODEE news-
letter, or Section 5.5 in the textbook
Differential Equations, A Modeling
Perspective, by B. Borrelli and
C. Coleman (1998, J. Wiley). 
----------------------------------------------
Project: find equilibria, study their stability and bifurcations
under different conditions
1) Show that nontrivial equilibria (endemic infection level) exists only for c < 1
2) Fix d=1, find and descride the bifurcations of equilibria in terms of parameter c

Try these combinations of parameters to
see other kinds of behavior:

1. c = 0, r = 2
2. c = 1.5, r = 2
3. c = .25, r = 2