Here we solve the optimal control problem using the indirect method [

35,

36]. This method has advantages and disadvantages. It is straightforward to implement if the numerical integrator uses a uniform step size and it has a high accuracy if a high order integrator is used together with a small uniform step size [

24,

32]. On the other hand, the main disadvantage is that it may be difficult to obtain convergence [

24]. However, solving the equations for the states and the adjoint variables as a two-point boundary value problem usually helps.

We use the bvp routine implemented in

python for solving the optimal control problem numerically [

37]. This routine uses a fourth order collocation algorithm with the control of residuals similar to the algorithm presented in [

38]. The produced collocation system is solved by a damped Newton method with an affine-invariant criterion function [

39]. The results were also obtained by using the forward-backward method which is well-known in optimal control literature [

24,

40].

We divide the presentation of the numerical results into two subsections for the sake of clarity. We start with the effects of the time-dependent control ${u}_{1}\left(t\right)$, which represents mass educational campaigns as explained before.

#### 4.1. Mass Educational Campaigns Effects

In this scenario, we consider the case where only the educational campaign policies are implemented as a control policy to minimize the spread of the Zika virus. As we have mentioned before, depending on the viewpoint, we might want to use a different functional

$\mathcal{J}$ (

4), or even different weights. Several ways can be used to assign the weights depending on costs of each control policy and the cost to health institutions and society of having infected people. For instance, whether infected people might need to see a doctor or occupy a room in a hospital. Thus, here we use a variety of values for the weights

${\kappa}_{i}$ in order to explore different impacts. We would like to remark that this is an important contribution related to the optimal control that helps with the analysis of the impact of the weights on the solution. Notice that the minimum of the control problem does not change if all the weights are multiplied by a constant. There are infinitely many options from which to choose these weights, but in the real-world, the weights must be assigned based on the priorities of the health institutions and on the costs associated to having an infected and symptomatic individual. In addition, it is important to take into account the costs of each control policy [

35,

41,

42,

43]; for example, the cost of educational campaigns. We will use metrics to measure the impact of each time-dependent control

${u}_{1}\left(t\right)$ and

${u}_{2}\left(t\right)$. The first metric is the total cases avoided by the control strategy during the full time period

T [

44]. This metric is computed using the following equation

where

${I}_{h}\left(0\right)$ is the initial number of infected persons and

${I}_{h}^{*}$ is the infected individuals corresponding to the optimal solution associated with the optimal controls. The second metric is given by:

This second metric is the effectiveness, which is the proportion of the total of avoided cases as compared to the total possible cases under a no intervention attitude [

44]. The efficacy function [

44] is defined by

For this first scenario we choose

${\kappa}_{1}=0.1$,

${\kappa}_{3}=0.1\phantom{\rule{0.166667em}{0ex}}{N}_{h}$, and

${\kappa}_{4}=0$. This means that we are only interested in minimizing the infected people, and using repellent, insect bednets, and appropriate clothing to avoid the bite of mosquitoes, i.e., just control

${u}_{1}$. We choose this particular set in such a way that the terms in the functional

$\mathcal{J}$ have a similar order of magnitude, and therefore the weights have impact on the functional

$\mathcal{J}$. The functional is given by

In this study, we assume that the maximum potential value for the control

${u}_{1}$ related to the educational campaign is

${u}_{1max}=0.5$. This is a particular value that can be changed in a real situation, and depends on the potential efficacy of the control. Notice that assuming

${u}_{1max}=1$ would give a potential 100% efficacy of the control

${u}_{1}$ and therefore no transmission of the Zika virus would happen. We think that this particular situation would be very optimistic for the real world. Therefore we used a more conservative efficacy of 50% (

${u}_{1max}=0.5$). In

Figure 1, the behavior of the infected population with and without control can be seen. In addition, the time-dependent control function

${u}_{1}\left(t\right)$ can be observed. We can also see that the implementation of a mass educational campaign would result in a reduction of the number of the infected population at any time. Notice, therefore, that the control stays at its maximum almost during the whole time period and that the effectiveness is not close to 100% since it cannot avoid all the infected cases. This makes sense since the educational campaign will not reach everybody right away. Furthermore, the maximum value that the control

${u}_{1}$ takes on during the simulation period is 0.5. In fact, this really means that the use of repellent, insect bednets, and appropriate clothing is necessary in order to have the optimal strategy to reduce the infected people.

The next variation that we include is to choose that the maximum potential value of the control of

${u}_{1}$ is

${u}_{1max}=0.05$ (efficacy of 5%). We can consider this as the population not being able to effectively grasp the educational campaign, and therefore the control will have less impact on avoiding the mosquito bites. Here we expect to have less effectiveness and averted cases. In

Figure 2, the infected population with and without control can be seen. We can see that there is a reduction of the number of infected population, but much less than in the previous case, since the control has a maximum value of

${u}_{1max}=0.05$. Notice that again the control stays at its maximum almost during the whole time period. This means that the use of repellent, insect bednets, and appropriate clothing is necessary in order to have the optimal strategy to reduce the number of infected persons.

Now we include the scenario where the cost of control

${u}_{1}$ is reduced to

${\kappa}_{3}=0.05\phantom{\rule{0.166667em}{0ex}}{N}_{h}$, i.e., half of the previous value. We kept the maximum value of the control of

${u}_{1}$ as

${u}_{1max}=0.05$. Since the cost associated to this control

${u}_{1}$ is lower, we expect to have more averted cases and effectiveness. In

Figure 3, the infected population with and without control can be seen. Notice that control

${u}_{1}$ stays at its maximum almost during the whole time period, and the number of averted cases increased due to lower cost of this control. This situation in the real world means that, if for instance the cost of the educational campaign (TV, radio, internet, use of repellent, insect bednets, and appropriate clothing) is reduced then the optimal strategy will reduce even more the number of infected individuals.

#### 4.2. Insecticide Spraying Campaign

Here, we consider that only the insecticide spraying campaign is implemented as a control policy to minimize the spread of the Zika virus by reducing the mosquito population choosing

${\kappa}_{1}=0.1$,

${\kappa}_{3}=0$, and

${\kappa}_{4}\ne 0$. We will minimize the following functional

First we choose

${\kappa}_{4}=0.1\phantom{\rule{0.166667em}{0ex}}{N}_{h}$, which is the same associated cost that we initially chose for the mass education campaign. In

Figure 4, the trajectories of the infected population with and without control can be seen. It is clear that the insecticide control strategy produces a reduction of the infected population and show the effectiveness of control

${u}_{2}\left(t\right)$. This time-dependent control disappears after around 100 days, since there are no infected mosquitoes at that time. Surprisingly, in this case we get better results as compared to the ones with the mass education control policy since we get more averted cases and furthermore the infected subpopulations disappear. Moreover, the control

${u}_{2}\left(t\right)$ never reaches the maximum value (

${u}_{2max}=0.5$) decreases over time, which means that the optimal strategy is to gradually diminish the use of insecticide over the time period. The decision of which control policy would be better might depend on the costs of the mass education (repellent, insect bednets, and appropriate clothing) and application of insecticide controls. In addition, it would depend on the efficacy of the insecticide. Here we kept it at

${u}_{2max}=0.5$, but in the real-world situation it may be different.

Now we will consider that the maximum value of the control of ${u}_{2}$ is ${u}_{2max}=0.05$, i.e., we reduce the insecticide killing efficacy on mosquitoes. Surprisingly, in this case there is no change since the weights are the same and in the previous case the maximum optimal value reached by control ${u}_{2}$ was 0.05. However, if we reduce the ${u}_{2max}=0.01$ the total averted cases decreases to 277,523 and the effectiveness to 0.53. In this particular case the time-dependent control ${u}_{2}\left(t\right)$ needs to be maintained for a longer time due to the reduction of the spraying efficacy on the killing of the mosquitoes.

The last case that we will consider with only insecticide control is lowering the associated cost of this control. Here we assume that

${\kappa}_{4}=0.01\phantom{\rule{0.166667em}{0ex}}{N}_{h}$ and

${u}_{2max}=0.5$. In

Figure 5, we can see (as we expected) that the insecticide control strategy substantially reduces the infected populations, and shows the effectiveness of control

${u}_{2}\left(t\right)$. This time-dependent control disappears after about 60 days, since there are no infected mosquitoes at that time. Therefore, no more insecticide is necessary. In a real world situation we might have an infected person coming from another place and a new Zika outbreak might start. The number of averted cases is 433,156, and the effectiveness reaches a value of 0.83. These effects are understandable since the cost associated to control

${u}_{2}\left(t\right)$ has been reduced and, therefore, the resources devoted to the implementation of control policies yield better performance.

#### 4.3. Mixing the Controls and Including the Infected Mosquitoes

We assume that the cost associated to one infected person is one fold more than the one associated to one infected mosquito. For this scenario we choose

${\kappa}_{1}=0.1$,

${\kappa}_{2}=0.01$, and

${u}_{1max}={u}_{2max}=0.5$. This situation means that we want to reduce both infected populations, i.e.,

${I}_{h},{I}_{M}$. The functional, similar to the one in (

4), is given by

First we only consider mass education control (to use of repellent, insect bednets, and appropriate clothing), i.e.,

${\kappa}_{4}=0$. As can be seen on the left-side of

Figure 6, control

${u}_{1}$ is used almost during the whole time period in order to reduce the infected population. However, the reduction is lower than in comparison when only the human infected population is included in the functional to be minimized. In fact, the averted cases are 260,096 and the effectiveness is 0.5. Then we did a simulation to consider only an insecticide campaign (

${\kappa}_{3}=0$). Surprisingly, the number of averted cases increased dramatically to 365,464 and the effectiveness to 0.7. Moreover, these results are better than in comparison to when only the human infected population is included in the functional to minimize. On the right-side of

Figure 6 it can be seen that control

${u}_{2}$ (insecticide) is only necessary for a little bit more than 100 days, and at a maximum optimal value of 0.05. In other words, the time-dependent control related to insecticide seems more efficient to decrease the spread of Zika virus. In this particular case the mosquito population survived, which can be seen as a good outcome from an ecological and biological viewpoint. It is important to remark that here we assume that

${\kappa}_{4}=0.1\phantom{\rule{0.166667em}{0ex}}{N}_{h}$, which is the associated cost related to the insecticide control

${u}_{2}$. This value is higher than the one related to infected persons, and that is a factor to explain why the optimal control strategy reduces the insecticide control

${u}_{2}$ faster in order to reduce the costs. Nevertheless, in the real world these associated cost values might vary due to different factors.

One last simulation that we would like to present in this subsection is to consider both controls with the same associated cost, i.e.,

${\kappa}_{3}={\kappa}_{4}$, and with the same amount of resources as in the previous case. Thus, we set

${\kappa}_{1}=0.09,$ and

${\kappa}_{2}=0.01$, in a such way that

${\sum}_{i=1}^{2}{\kappa}_{i}=0.1$. In regard to the costs of the controls we split resources assigning values

${\kappa}_{3}={\kappa}_{4}=0.05\phantom{\rule{0.166667em}{0ex}}{N}_{h}$. The simulation of this last scenario, provides an important example of how synergy works for the two controls. Here we obtained better results (same resources) than in comparison with previous cases, where one control was used without the other and vice-versa. The number of averted cases is 384,578 and the effectiveness is 0.74. In

Figure 7, it can be seen that both infected populations are reduced and eliminated. Thus, we can conclude that a combination of both controls is the best strategy assuming that the cost associated to one infected person is one fold greater than having one infected mosquito. In the real world, the cost associated to infected people is much higher than the one related to infected mosquitoes. However, we have seen that including the infected mosquito population in the functional is a good strategy to reduce the Zika virus spread.