Table 2.

Transition probability fluxes for Ross-Macdonald model. Numbers of infectious humans and populations are denoted by X=(H, M)^T. The vector ΔX=(ΔX ⁱ)=(H(t+Δt)−H(t), M(t+Δt)−M(t))^T denotes change in state, i=1, 2, …, 5

Event i	Change in population sizes	(ΔX i)	Transition probability p i
Infection of susceptible humans	$(\mathit{H},\mathit{M})\to (\mathit{H}+1,\mathit{M})$	$\left(\begin{array}{l}1\\ 0\end{array}\right)$	${\mathit{p}}_1=\frac{\mathit{k}\left(\mathit{t}\right)\mathit{pM}({\mathit{N}}_{\mathit{h}}-\mathit{H})}{{\mathit{N}}_{\mathit{h}}}\mathrm{\Delta t}$
Recovery of infectious humans	$(\mathit{H},\mathit{M})\to (\mathit{H}-1,\mathit{M})$	$\left(\begin{array}{l}-1\\ 0\end{array}\right)$	p 2 = μ(t)HΔt
Infection of susceptible mosquitoes	$(\mathit{H},\mathit{M})\to (\mathit{H},\mathit{M}+1)$	$\left(\begin{array}{l}0\\ 1\end{array}\right)$	${\mathit{p}}_3=\frac{\mathit{k}\left(\mathit{t}\right)\mathit{qH}\left({\mathit{N}}_{\mathit{m}}\right(\mathit{t})-\mathit{M})}{{\mathit{N}}_{\mathit{h}}}\mathrm{\Delta t}$
Death of infectious mosquitoes	$(\mathit{H},\mathit{M})\to (\mathit{H},\mathit{M}-1)$	$\left(\begin{array}{l}0\\ -1\end{array}\right)$	p 4 = δ(t)MΔt
No change	$(\mathit{H},\mathit{M})\to (\mathit{H},\mathit{M})$	$\left(\begin{array}{l}0\\ 0\end{array}\right)$	$1-\sum _{\mathit{i}=1}^4{\mathit{p}}_{\mathit{i}}$

Transitions are presented in their most general form by expressing parameters that may be influenced by control measures as functions of time