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, ($\lambda =\mathrm{\Lambda }/H$), then $$\frac{dm}{dt}=\lambda -gm$$ and at the steady state, $$\overline{m}=\frac{\lambda }{g}$$

If we substitute this $\overline{m}$ 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:

$$R_0=bV\frac{c}{r}$$