Table 1.

Variables and parameters of the time-varying Ross-Macdonald models

Variable	Expression	Value
Number of infectious humans	H
Number of infectious mosquitoes	M
Per-capita human recovery rate	μ	0.01 day−1 (Smith and McKenzie 2004)
Per-capita mosquito mortality rate	δ	0.1 day−1 (Smith and McKenzie 2004)
Human population size	N h	1000
Per-capita mosquito biting rate	k	0.3 day−1 (Smith and McKenzie 2004)
Transmission efficiency from mosquitoes to humans	p	0.5 (Smith and McKenzie 2004)
Transmission efficiency from humans to mosquitoes	q	0.5 (Smith and McKenzie 2004)
Mosquito population size	N m	10000
Basic reproduction number	${\mathit{R}}_0=\frac{{\mathit{k}}^2\mathit{pq}{\mathit{N}}_{\mathit{m}}}{\mathit{\delta \mu }{\mathit{N}}_{\mathit{h}}}$
Time critical point is reached	t ∗
Critical biting rate	${\mathit{k}}^{\ast }=\sqrt{\left({\mathit{N}}_{\mathit{h}}\mathit{\delta \mu }\right)/\left(\mathit{pq}{\mathit{N}}_{\mathit{m}}\right)}$
Critical mosquito population size	${\mathit{N}}_{\mathit{m}}^{\ast }=\left({\mathit{N}}_{\mathit{h}}\mathit{\delta \mu }\right)/\left(\mathit{pq}{\mathit{k}}^2\right)$
Critical recovery rate	μ ∗=(k 2 p q N m)/(N h δ)
Critical mortality rate	δ ∗=(k 2 p q N m)/(N h μ)
Value of biting rate prior to application of control measures	k 0	0.3 day−1
Rate of change in biting rate	k 1	0.0001 day−1
Mosquito population size prior to application of control measures	N m0	10000
Rate of change of mosquito population size	N m1	1 day−1
Mosquito mortality rate prior to application of control measures	δ 0	0.1 day−1
Rate of change of mosquito mortality rate	δ 1	0.0025 day−1
Value of human recovery rate prior to application of control measures	μ 0	0.01 day−1
Rate of change of recovery rate	μ 1	0.001 day−1
Time-varying biting rate	$\mathit{k}\left(\mathit{t}\right)=\left\{\begin{array}{ll}{\mathit{k}}_0-{\mathit{k}}_1\mathit{t},& \mathit{t}<{\mathit{t}}^{\ast }\\ {\mathit{k}}^{\ast }& \mathit{t}\ge {\mathit{t}}^{\ast }\end{array}\right.$	t ∗=2800 days
Time-varying mosquito population size	${\mathit{N}}_{\mathit{m}}\left(\mathit{t}\right)=\left\{\begin{array}{ll}{\mathit{N}}_{\mathit{m}0}-{\mathit{N}}_{\mathit{m}1}\mathit{t},& \mathit{t}<{\mathit{t}}^{\ast }\\ {\mathit{N}}_{\mathit{m}}^{\ast }& \mathit{t}\ge {\mathit{t}}^{\ast }\end{array}\right.$	t ∗=9956 days
Time-varying mortality rate	$\mathit{\delta }\left(\mathit{t}\right)=\left\{\begin{array}{ll}{\mathit{\delta }}_0+{\mathit{\delta }}_1\mathit{t},& \mathit{t}<{\mathit{t}}^{\ast }\\ {\mathit{\delta }}^{\ast }& \mathit{t}\ge {\mathit{t}}^{\ast }\end{array}\right.$	t ∗=8960 days
Time-varying recovery rate	$\mathit{\mu }\left(\mathit{t}\right)=\left\{\begin{array}{ll}{\mathit{\mu }}_0+{\mathit{\mu }}_1\mathit{t},& \mathit{t}<{\mathit{t}}^{\ast }\\ {\mathit{\mu }}^{\ast }& \mathit{t}\ge {\mathit{t}}^{\ast }\end{array}\right.$	t ∗=2240 days
Environmental noise strength	σ	0.05