## 1. Introduction

## 2. Models Formulation

#### 2.1. The Basic System

Parameter | Description | Baseline Value |
---|---|---|

${\Lambda}_{h}$ | Immigration rate in humans | ${10}^{3}/(70\times 365)$ |

${\Lambda}_{v}$ | Immigration rate in mosquitoes | ${10}^{4}/21$ |

b | Proportion of ITN usage | varies in [0,1] |

μ | Natural mortality rate in humans | $1/(70\times 365)$ |

η | Natural mortality rate in mosquitoes | $1/21$ |

${\eta}_{bn}$ | Maximum ITN-induced death rate in mosquitoes | $1/21$ |

α | Disease–induced death rate in humans | ${10}^{-3}$ |

${p}_{1}$ | Prob. of disease transm. from mosquito to human | 1 |

${p}_{2}$ | Prob. of disease transm. from human to mosquito | 1 |

${\beta}_{max}$ | Maximum host–vector contact rate | 0.1 |

${\beta}_{min}$ | Minimum host–vector contact rate | 0 |

δ | Recovery rate of infectious humans to be susceptible | 1/4 |

- (i)
- ITNs usage reduces the human–mosquito contact rate and this is described by the relation:$$\beta \left(b\right)={\beta}_{max}-b\left({\beta}_{max}-{\beta}_{min}\right),$$
- (ii)
- ITNs usage increases the mosquito death rate η. This is modeled by:$$\eta \left(b\right)=\eta +{\eta}_{bn}b,$$

#### 2.2. Information Variable

- (a)
- the information on the status of the disease in the community where they live;
- (b)
- the information generated by an optimal health-promotion campaign aimed at encouraging people to use ITNs.

#### 2.3. The Information Kernel

#### 2.4. The Cases under Consideration

- Case I (
**Model M**): Constant contact rate.

- Case II (
**Model V**): Pure voluntary ITNs adoptions.

- Case III (
**Model P**): ITNs promotion program.

**Model P**:

## 3. Optimal ITNs Promotion Campaign

## 4. Numerical Simulations and Discussion

**Figure 1.**Panel A: Dynamics of infectious humans ${I}_{h}$ and infectious vector ${I}_{v}$ as predicted by Model M (9). The parameter values are given in Table 1, except that $\beta =0.09$ and $\eta =0.0524$. Initial data are $({N}_{h0},{I}_{h0},{S}_{v0},{I}_{v0})=(910,10,4000,1000)$. Panel B: Dynamics of ${I}_{h}$ and ${I}_{v}$ as predicted by Model V (12) with parameter values given in Table 1, except that $\beta =0.09$ and $\eta =0.0524$. Furthermore, $\xi =0.01$, $\gamma =0.01$, ${w}_{max}=1$. The initial data are $({N}_{h0},{I}_{h0},{S}_{v0},{I}_{v0},{w}_{0})=(910,10,4000,1000,0)$.

**Figure 2.**Predictions by Model M, Model V and by the optimal strategy implemented on Model P. Panel A: Infectious humans. Panel B: contact rate. We use the parameter values given in Table 1, except that $\beta =0.09$ and $\eta =0.0524$. Furthermore, $\xi =0.01$, $\gamma =0.01$. The cost weights are $A=1$, $B=10$. The initial data are $({N}_{h0},{I}_{h0},{S}_{v0},{I}_{v0},{w}_{0})=(910,10,4000,1000,0)$.

**Figure 3.**Panel A: The optimal control profile. Panel B: Dynamics of infectious humans ${I}_{h}$ and infectious vector ${I}_{v}$ as output of the optimal control problem given by Model P (14) and the minimization of the objective functional (15). The parameter values are those specified in the captions of Figure 1 and Figure 2.

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix

## Optimality System and Basic Properties of the Optimal Control

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