Background: The sterile insect technique and transgenic equivalents are considered promising tools for controlling vector-borne disease in an age of increasing insecticide and drug-resistance. Combining vector interventions with artemisinin-based therapies may achieve the twin goals of suppressing malaria endemicity while managing artemisinin resistance. While the cost-effectiveness of these controls has been investigated independently, their combined usage has not been dynamically optimized in response to ecological and epidemiological processes.

Results: An optimal control framework based on coupled models of mosquito population dynamics and malaria epidemiology is used to investigate the cost-effectiveness of combining vector control with drug therapies in homogeneous environments with and without vector migration. The costs of endemic malaria are weighed against the costs of administering artemisinin therapies and releasing modified mosquitoes using various cost structures. Larval density dependence is shown to reduce the cost-effectiveness of conventional sterile insect releases compared with transgenic mosquitoes with a late-acting lethal gene. Using drug treatments can reduce the critical vector control release ratio necessary to cause disease fadeout.

Conclusions: Combining vector control and drug therapies is the most effective and efficient use of resources, and using optimized implementation strategies can substantially reduce costs.

Keywords: ACT; Artemisinin; Cost-effectiveness; Malaria management; Optimal control; Vector control.

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**Fig. 1**
Vector control and drug therapies are most effective when used in tandem.?Proportions of humans (h) and vectors (v) infected are plotted under four control scenarios. a no control, $w = 0$ , $u = 0$ ; b only artemisinin treatment, $w = 0.05$ ( $5 %$ drug coverage), $u = 0$ ; c only vector control, $w = 0$ , $u = 0.2$ (releasing modified males at a rate of $20 %$ of the wild male populations per day); d both artemisinin treatment and vector control, $w = 0.05$ , $u = 0.2$ . The total number of infected mosquitoes at time $t = 90$ days for each scenario is a 6125 b 1145 c 238 d 55; the vector control suppresses the vector population significantly. Early-acting SIT is assumed

**Fig. 2**
Vector control broadens disease-free parameter space, causing disease fade-out even for high biting rates. The level of endemic disease, $h^{*}$ , across b–w parameter space (biting rate–drug treatment proportion) with a no vector control, b early-acting SIT releases and c late-acting SIT releases. The release ratio in b and c is $u = 0.1$ (daily releases of $10 %$ of the wild male population), with the ratio being of the current vector population A(t) rather than the equilibrium population $A^{*}$ (elsewhere in the paper the ratio of $A^{*}$ is used). Disease-free regions of parameter space are coloured white. Other parameters as in Table 1

**Fig. 3**
Vector control releases can achieve disease fade-out for sparse drug coverage when biting rates are below $b < 1.2$ . The critical release ratio $u = u_{c}$ that leads to disease fadeout ( $h^{*} \to 0$ ) for a given treatment proportion w as the mosquito biting rate b varies, as calculated using (46) and (47). The release ratio u is of the current vector population A(t) rather than the equilibrium population $A^{*}$ (elsewhere in the paper the ratio of $A^{*}$ is used). Early and late SIT are compared; all other parameters as in Table 1

**Fig. 4**
Optimal strategies for releases and drug treatments can substantially reduce the cost of managing malaria. a The proportion of infected humans under six control scenarios (where w is the host treatment proportion and u is the insect release ratio with respect to the wild vector population): no control (x-axis label “0,0”), $w = w_{0}$ , $u = 0$ (x-axis label “ $w_{0}$ ,0”), $w = 0$ , $u = u_{0}$ (x-axis label “0, $u_{0}$ ”), $w = w_{0}$ , $u = u_{0}$ (x-axis label “ $w_{0}$ , $u_{0}$ ”), $w = 0$ , $u = u^{*} (t)$ (x-axis label “0, $u^{*}$ ”), $w = w^{*} (t)$ , $u = u^{*} (t)$ (x-axis label “ $w^{*}$ , $u^{*}$ ”), where $w_{0} = 0.05$ , $u_{0} = 0.2$ ( $5 %$ drug coverage and releasing $20 %$ of the wild male population per day) and $w^{*}$ , $u^{*}$ are the optimal control strategies defined in (18) and (29), respectively. b The total cost of the scenario, including spending on traditional healthcare (h), spending on artemisinin treatment (w) and spending on insect releases (u). Early-acting SIT is assumed. Quadratic cost functions of h, w and u are assumed

**Fig. 5**
Late-acting lethality suppresses the effects of over-compensatory larval density dependence. a Cost per case averted (excluding initial capital investment in construction) for optimized release strategies of early-acting and late-acting SIT, for the parameters given in Table 1, as the strength of the larval density dependence $β$ is increased. The density dependence runs from contest for $β < 1$ to scramble for $β > 1$ . The number of cases averted is approximated by ([mean no. with no control???mean no. with control]/average duration of disease) $\times$ no. of days). The change in the optimal release strategy is shown for b early SIT, and c late SIT for three specific values of $β$ . Artemisinin treatment is not used

**Fig. 6**
Vector control costs, unlike medical care costs, are insensitive to the form of cost function. a Cost per capita for the optimal strategy over a 40 day control period for three cost functions: linear, quadratic and square root; and three epidemiological parameter sets: low, medium and high (for low $a, b, c, k^{*} = 75 %$ of medium, for high $a, b, c, k^{*} = 125 %$ of medium). In the top figure of a the cost function $C_{h}$ is varied while keeping $C_{u}$ and $C_{w}$ quadratic ( $m_{2} = m_{3} = 2$ ); in the bottom figure the cost function $C_{u}$ is varied while keeping $C_{h}$ and $C_{w}$ quadratic ( $m_{1} = m_{2} = 2$ ). The number of cases averted is approximated by ([mean no. with no control - mean no. with control]/average duration of disease) $\times$ no. of days). b A sketch of the three cost functions ${\hat{θ}}_{i} (\cdot)$ (linear), ${\hat{θ}}_{i} {(\cdot)}^{1 / 2}$ (square root) and ${\hat{θ}}_{i} {(\cdot)}^{2}$ (quadratic), where the altered price ${\hat{θ}}_{i}$ that multiplies the cost function is: 0.8 (square root), 1 (linear), 1.5 (quadratic). Late-acting SIT is used

**Fig. 7**
Increasing the initial insect release ratio has diminishing returns on the effect on disease prevalence. Example solutions from the default epidemiological parameter set of Fig. 6a when $C_{u}$ is varied through square root, linear and quadratic. a Shows the optimal vector release strategy, b shows the resulting suppression of disease in the host population. Late-acting SIT is used. Other parameters as in Table 1

**Fig. 8**
Migration can harm concentrated control efforts; distributed vector control with combined drug treatment is optimal. The proportion of hosts infected in each of three identical populations with SIT release ratios $u_{1}$ , $u_{2}$ and $u_{3}$ , where migration is $10 %$ between nodes except in a where there is no migration. a $u_{1} = 3$ , $u_{2} = u_{3} = w_{i} = 0$ ; b $u_{1} = 3$ , $u_{2} = u_{3} = w_{i} = 0$ (concentrated control); c $u_{1} = u_{2} = u_{3} = 1$ . $w_{i} = 0$ (distributed control); d $u_{i} (t) = u_{i}^{*} (t)$ , $w_{i} = 0$ (optimal vector control); e $u_{i} (t) = u^{*} (t)$ , $w_{i} (t) = w_{i}^{*} (t)$ (optimal combined drug treatment and vector control). f Cost breakdowns for the five scenarios plotted above. All parameters as in Table 1. Late-acting lethality is used

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Optimal control of malaria: combining vector interventions and drug therapies

Abstract

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