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Transcript
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