Download Calculating the optimal human density, σ*

Calculating the optimal human density, σ*
As stated, ‘optimal human density’ refers to the density of humans at which R0 is
maximized. We therefore start with the equation for R0:
me  T
(r )G 2
R0 
Equation A1
We separate the elements which are affected by human density from the elements which
are not:
R0 
m e  T

G 2 r
Equation A2
Hence:
R0 
m
G2
Equation A3
Higher human density (σ) decreases m by a factor of 1/σ, therefore in order to maximize
R0, we want to maximize the expression:
1
 G2
Expression A4
Beginning with the situation whereby  
1
1 

  

s 

1
, we want to maximize:
s
Expression A5
2
Multiplying the brackets out results in:
1
2
1
 2 
 2
s
s
Expression A6
Maximizing Expression 6 is the same as minimizing its inverse. To do this, we have to
find the derivative of the inverse of Expression 6, set it to equal zero and solve for σ:
f ( )   2 
f ' ( )   2 
2
1
 2
s s
1
s 2
2
Equation A7
Equation A8
0 2 
1
Equation A9
s 2
Which becomes:
1
* 
s
2
Finally, substituting
* 
Equation A10
1
1

for ε we can solve for the condition that
s
s
1
1
 
1
1
s
s 
Hence, we show that:
1
1
 s , if   s
*  
1
  , if  
s

Equation A11
Equation A12