Smith & McKenzie

Basic Dynamics

Smith & McKenzie

With a few small modifications, we can begin to understand Macdonald’s formulas in a way that tells the story of transmission. Smith and McKenzie split parameter describing human blood feeding into:

• \(f\) – the overall blood feeding rate

• \(q\) – the human fraction

such that \(a=fq.\) and \[S = \frac{fq}g\]

Let \(\lambda\) denote the scaled emergence rate of mosquitoes, (\(\lambda = \Lambda/H\)), then \[\frac{dm}{dt} = \lambda - g m \] and at the steady state, \[\bar m = \frac \lambda g\]

If we substitute this \(\bar m\) into the expression for vectorial capacity, we get

\[V = \lambda \frac{f^2 q^2}{g^2} e^{-g\tau} = \lambda S^2 e^{-g\tau}\]

The right hand side of the equation tells the story of transmission by mosquitoes: after emerging (\(\lambda\)), a mosquito must blood feed on a human to become infected (the first \(S\)), survive the EIP (\(e^{-g\tau}\)), and then blood feed on humans to transmit (the second \(S\)).

The expression for \(R_0\) is simply:

\[R_0 = b V \frac cr\]