Impulsive release strategies of sterile mosquitos for optimal control of wild population

Figures & data

Table 1. Value of model parameters.

Parameter	Value	Range	Unit	Source
$\overline{r}$	14	7.46−14.85	day${}^{-1}$	[30]
$\overline{s}$	0.5	0.5	1	[14]
α	6000	6000	1	Assumed
γ	12,000	12000	1	Assumed
$d_0$	0.04	0.034−0.05	day${}^{-1}$	[30]
$d_1$	0.003	0.003	day${}^{-1}$	Assumed
${\mu }_1$	0.046	0.033−0.046 (Female)	day${}^{-1}$	[30]
${\mu }_1$	0.046	0.077−0.139 (Male)	day${}^{-1}$	[30]
${\mu }_2$	0.04	0.04	day${}^{-1}$	[10]

Figure 1. The global stability of the boundary periodic solution of system (2(2) $\left\{\begin{array}{l}\left.\begin{array}{l}\frac{\mathrm{d}J\left(t\right)}{\mathrm{d}t}=\frac{rA\left(t\right)}{A\left(t\right)+g\left(t\right)}A\left(t\right)-\frac{\alpha J\left(t\right)}{\gamma +J\left(t\right)}\\ -(d_0+d_{1}J(t\left)\right)J\left(t\right),\\ \frac{\mathrm{d}A\left(t\right)}{\mathrm{d}t}=\frac{\alpha J\left(t\right)}{\gamma +J\left(t\right)}-{\mu }_{1}A\left(t\right),\\ \frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}=-{\mu }_{2}g\left(t\right),\end{array}\right\}t\ne kp,k=1,2,\dots ,\\ \left.\begin{array}{l}J\left(t^{+}\right)=J\left(t\right),\\ A\left(t^{+}\right)=A\left(t\right),\\ g\left(t^{+}\right)=g\left(t\right)+\sigma ,\end{array}\right\}t=kp\end{array}\right.$(2) ).

Figure 2. System (2(2) $\left\{\begin{array}{l}\left.\begin{array}{l}\frac{\mathrm{d}J\left(t\right)}{\mathrm{d}t}=\frac{rA\left(t\right)}{A\left(t\right)+g\left(t\right)}A\left(t\right)-\frac{\alpha J\left(t\right)}{\gamma +J\left(t\right)}\\ -(d_0+d_{1}J(t\left)\right)J\left(t\right),\\ \frac{\mathrm{d}A\left(t\right)}{\mathrm{d}t}=\frac{\alpha J\left(t\right)}{\gamma +J\left(t\right)}-{\mu }_{1}A\left(t\right),\\ \frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}=-{\mu }_{2}g\left(t\right),\end{array}\right\}t\ne kp,k=1,2,\dots ,\\ \left.\begin{array}{l}J\left(t^{+}\right)=J\left(t\right),\\ A\left(t^{+}\right)=A\left(t\right),\\ g\left(t^{+}\right)=g\left(t\right)+\sigma ,\end{array}\right\}t=kp\end{array}\right.$(2) ) has two locally stable periodic solutions with $\sigma =22,000<{\sigma }^{\ast }$: a positive coexistence one and a boundary one.

Figure 3. System (2(2) $\left\{\begin{array}{l}\left.\begin{array}{l}\frac{\mathrm{d}J\left(t\right)}{\mathrm{d}t}=\frac{rA\left(t\right)}{A\left(t\right)+g\left(t\right)}A\left(t\right)-\frac{\alpha J\left(t\right)}{\gamma +J\left(t\right)}\\ -(d_0+d_{1}J(t\left)\right)J\left(t\right),\\ \frac{\mathrm{d}A\left(t\right)}{\mathrm{d}t}=\frac{\alpha J\left(t\right)}{\gamma +J\left(t\right)}-{\mu }_{1}A\left(t\right),\\ \frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}=-{\mu }_{2}g\left(t\right),\end{array}\right\}t\ne kp,k=1,2,\dots ,\\ \left.\begin{array}{l}J\left(t^{+}\right)=J\left(t\right),\\ A\left(t^{+}\right)=A\left(t\right),\\ g\left(t^{+}\right)=g\left(t\right)+\sigma ,\end{array}\right\}t=kp\end{array}\right.$(2) ) has two locally stable periodic solutions with $p=3>p^{\ast }$: a positive coexistence one and a boundary one.

Figure 4. The global stability of the boundary periodic solution of system (2(2) $\left\{\begin{array}{l}\left.\begin{array}{l}\frac{\mathrm{d}J\left(t\right)}{\mathrm{d}t}=\frac{rA\left(t\right)}{A\left(t\right)+g\left(t\right)}A\left(t\right)-\frac{\alpha J\left(t\right)}{\gamma +J\left(t\right)}\\ -(d_0+d_{1}J(t\left)\right)J\left(t\right),\\ \frac{\mathrm{d}A\left(t\right)}{\mathrm{d}t}=\frac{\alpha J\left(t\right)}{\gamma +J\left(t\right)}-{\mu }_{1}A\left(t\right),\\ \frac{\mathrm{d}g\left(t\right)}{\mathrm{d}t}=-{\mu }_{2}g\left(t\right),\end{array}\right\}t\ne kp,k=1,2,\dots ,\\ \left.\begin{array}{l}J\left(t^{+}\right)=J\left(t\right),\\ A\left(t^{+}\right)=A\left(t\right),\\ g\left(t^{+}\right)=g\left(t\right)+\sigma ,\end{array}\right\}t=kp\end{array}\right.$(2) ) with $p=1.5<p^{\ast }$.

Figure 5. Release amount control: (a) Comparisons of total wild mosquitoes population under different biological controls; (b) Impact of the intensity of each release on the objective function and wild mosquito population at time T.

Figure 6. Release timing control: (a) Comparisons of total wild mosquitoes population under different biological controls; (b) Release strategy of the mixed optimal control; (c) Impact of the intensity of each release on the optimal cost value and wild mosquito population at time T; (d) Errors of the cost function $\mathcal{J}$ in each iteration for optimal release timing control.

Table 2. Values of the cost function in the iteration process for optimal release timing control.

Number of iterations	0	3	6	9	12
$\mathcal{J}(\times {10}^4)$	2.1556	1.9443	1.7638	1.6799	1.6510

Figure 7. Mixed control: (a) Comparisons of total wild mosquitoes population under different biological controls; (b) Comparisons of total release amounts of sterile mosquitoes for three optimal control methods; (c) Release strategy of the mixed optimal control; (d) Errors of the cost function $\mathcal{J}$ in each iteration for mixed optimal control.

Table 3. Values of the cost function in the iteration process for optimal mixed control.

Number of iterations	0	6	12	18	24
$\mathcal{J}(\times {10}^4)$	2.1556	1.8746	1.7132	1.6324	1.5965

Table 4. Comparison of different release strategies.

	Optimal control parameters	${\mathcal{J}}^{\ast }$	$J^{\ast }\left(T\right)+A^{\ast }\left(T\right)$
Release amount control	${\sigma }_f^{\ast }=57350$	20,550	15,962
	$T_1^{\ast }=0.0405$, $T_2^{\ast }=0.2126$,
Release timing control	$T_3^{\ast }=0.2126$, $T_4^{\ast }=4.5363$,	16,510	12,107
	$T_5^{\ast }=9.9980$, ${\sigma }_f^{\ast }=55,040$
	$T_1^{\ast }=0.0162$, $T_2^{\ast }=0.1025$,
	$T_3^{\ast }=0.2025$, $T_4^{\ast }=4.6789$,
Mixed control	$T_5^{\ast }=9.9999$, ${\sigma }_1^{\ast }=99,940$	15,965	11,965
	${\sigma }_2^{\ast }=99,940$, ${\sigma }_3^{\ast }=8640$
	${\sigma }_4^{\ast }=8640$

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