Agent-Based Models and Optimal Control in Biology: A Discrete Approach - Mathematical Concepts and Methods in Modern Biology - page 156

Since agent-based models are generally stochastic in nature, the data obtained will
seldom present a perfect description of the system, because an infinite number of
simulations would be required. Thus sample correlation coefficients of 1 or 1are

very highly unlikely. We may choose our desired correlation coefficient r , and when
scaling the model we simply select the smallest dimensions that produce data whose
correlation coefficient is at or above this level.

−

5.6.2 Cost Function Analysis When Scaling
Once we have determined how small we can safely scale our model, we must also
be cautious about numerical results obtained from this model, as indicated in the
following example.
Example 5.3. Let r i be the number of rabbits alive on day i , and let u i be the
control decision on day i (thus if we use poison then u i = 1 and if not, u i = 0). Let

u =[ u 1 u 2 ... u n ] , where n is the total number of simulated days. Suppose our cost

· i = 1 r i + · i = 1 u i , where a function is c are constants. Once

we have scaled our model, we attempt to use it to determine the control schedule
which minimizes this cost function. When doing so, we must be careful to scale the
constants a

( u ) = a b , b ∈ R

b as well, since otherwise, we may obtain meaningless or misleading
results, as the following results suggest.
Example 5.4. Let a

,

= 100 , b = 2000 , n = 5 , u =[ 01101 ] , v =[ 10000 ] . Suppose

the average rabbit numbers for and

by scaling, we are able to reduce the model size and achieve average rabbit numbers
{75, 15, 20, 27, 36}. Using another control schedule

u for the 5 simulated days are { 150 , 30 , 40 , 54 , 72 }

we obtain population levels
{150, 200, 40, 8, 10} (and the corresponding scaled population values are {75, 100,
20, 4, 5}).
Comparing the cost of the two schedules at the original size, we have:

v

( ) = ( + + + + ) + ( + + + + ) = ,

c u 100 150 30 40 54 72 2000 0 1 1 0 1 40600

c (v) = 100 ( 150 + 200 + 40 + 8 + 10 ) + 2000 ( 1 + 0 + 0 + 0 + 0 ) = 42800 .

Without scaling coefficients a , b , we would obtain the following costs using the

scaled model:

c ( u ) = 100 ( 75 + 15 + 20 + 27 + 36 ) + 2000 ( 0 + 1 + 1 + 0 + 1 ) = 23300 ,

c (v) = 100 ( 75 + 100 + 20 + 4 + 5 ) + 2000 ( 1 + 0 + 0 + 0 + 0 ) = 22400 .

From the original model, we conclude that (since the

associated cost is lower), but from the reduced model we conclude that in fact

u is a better solution than v

v is

better than u . This is due to the fact that when scaling the population values, even

though the dynamics correlated perfectly, the coefficients now give less weight to the
rabbit values, since these numbers are smaller in magnitude. In order to compensate
for this, we must account for the rabbit numbers being halved by doubling a .Atthe
same time, we need not scale b because the number of days remains constant at all