# The Role of Vertical Transmission in the Control of Dengue Fever

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## Abstract

**:**

## 1. Introduction

## 2. A Two-Strain Dengue Model with Control

## 3. An Optimal Control Problem

## 4. Numerical Results

#### 4.1. The Impact of the Relative Cost on the Controlled Dengue Dynamics

#### 4.2. The Impact of Vertical Transmission on the Controlled Dengue Dynamics

#### 4.3. The Impact of Vertical Transmission on the Objective Function

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Existence of Optimal Control

- The set of controls and corresponding state variables is non-empty.
- The control set, $\Omega ,$ is convex and closed.
- The right hand side of system (2) is bounded by a linear function in the state and control.
- The integrand of the objective functional is convex and bounded below by ${c}_{1}(|{u}_{1}{|}^{2}+|{u}_{2}{{|}^{2})}^{\frac{\beta}{2}}-{c}_{2}$ and the Lipschitz condition is satisfied.
- The payoff function is continuous.

**Proof**

- If we consider the vector of state variables $\mathit{x}\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}{[S,\phantom{\rule{0.222222em}{0ex}}{D}_{Am},\phantom{\rule{0.222222em}{0ex}}{D}_{As},\phantom{\rule{0.222222em}{0ex}}H,\phantom{\rule{0.222222em}{0ex}}R,\phantom{\rule{0.222222em}{0ex}}V,\phantom{\rule{0.222222em}{0ex}}{W}_{Am},\phantom{\rule{0.222222em}{0ex}}{W}_{As}]}^{T}$, then we can write our system of equations as$$\dot{\mathbf{x}}\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}f(\mathbf{x},u).$$Since we know our state variables are bounded in the positive orthant, the particular form of our system of equations dictates that $f(\mathit{x},u)$ is bounded. Thus there exists a unique solution to our system given suitable initial conditions.
- By the construction of $\Omega $, this condition is clearly met.
- The total population for both the host and vector systems is constant, thus all solutions are bounded. The control function is also bounded, thus the right-hand side can be bounded by a linear function in the state and control.
- The integrand is linear in the state variable and quadratic in the control function, and thus clearly convex. Furthermore, the Lipschitz is condition is clearly satisfied as the integrand is bounded below since both the state and control are non-negative.
- The payoff function is clearly continuous by construction.

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**Figure 1.**Host model flow diagram: S is the class of susceptible individuals who can become infectious with either DENV-2 American genotype, ${D}_{Am}$, or DENV-2 Asian genotype ${D}_{As}$ via infectious female mosquitoes W carrying the corresponding strain. In this model, only individuals infected with the Asian genotype can progress to DHF, H, and all infected individuals can recover, R. Note that the control function $(1-u(t)$) is modeled as the reduction efforts in the transmission rate from S either to ${D}_{Am}$, or ${D}_{As}$.

**Figure 2.**Vector model flow diagram: V is the class of susceptible female mosquitoes that can become infected with either DENV-2 American genotype ${W}_{Am}$ or DENV-2 Asian genotype ${W}_{As}$ via contact with an infectious human, D carrying the corresponding genotype. Vertical transmission only occurs in mosquitoes infected with genotype Asian. In this model, there is a constant birth rate, but a proportion, p, of those births by mosquitoes carrying genotype Asian, ${W}_{As}$, enter directly into the infectious class. Note that the control function $(1-u(t)$) is modeled as the reduction efforts in the transmission rate from V either to ${W}_{Am}$, or ${W}_{As}$.

**Figure 3.**As the relative cost of the control function, w

_{3}, is reduced, the proportion of infected people decreases. However when the outbreak is dominated by the strain without vertical transmission, (

**a**) then the outbreak can be controlled more easily than when the outbreak is dominated by the strain with vertical transmission, (

**b**) In the latter case, the cost of control must be reduced even further to effectively control the outbreak.

**Figure 4.**As the relative cost of the control function decreases, it is used more frequently and is able to control the outbreak. If the relative cost is expensive, then it is used sparingly and in response to outbreaks. Notice the peaks occur right after an increase in the prevalence of dengue in the corresponding panel of Figure 3.

**Figure 5.**When the dominant strain is DENV-2 American, as the level of vertical transmission increases, the level of optimal control increases (

**b**). Therefore, it is easier to control the outbreak (

**a**).

**Figure 6.**When the dominant strain is DENV-2 Asian, as the level of vertical transmission increases, the level of optimal control increases (

**b**). However, it is harder to control the outbreak (

**a**).

**Figure 7.**Even with an “effective” control program, a high vertical transmission rate can render the health policy moot regardless of the cost of additional control is low (

**a**), or high (

**b**).

**Figure 8.**The outbreak can be well controlled except when vertical transmission is extremely high. Although this situation is unrealistic, it highlights the importance of a control strategy, whether highly cost-efficient, left, or otherwise to take into consideration all possible transmission pathways.

**Figure 9.**Regardless of whether horizontal transmission is low, left panels, or moderate, right panels, a high level of vertical transmission can create extremely large, and costly outbreaks, top panels. If the relative costs of controlling the outbreak are low, w

_{3}, then the epidemic can still be controlled, bottom panels. However, if the cost is high, then the outbreak will be extremely expensive and impossible to control.

**Table 1.**Default Parameter Values: Biological parameters may vary across geographic and temporal scales, however, most of the values are taken from related literature or estimated to achieve the desired reproductive number.

Parameter | Default Value | Units | Source |
---|---|---|---|

M | 1 | per day | [52] |

N | 1 | per day | [52] |

$\alpha $ | 0.113 | per day | [52] |

${\mu}_{m}$ | 0.0958 | per day | [53] |

p | 0–1 | proportion | [27] |

$\mu $ | 0.00038 | per day | estimated |

${\theta}_{As}$ | 0.28 | per day | estimated |

$\theta Am$ | 0.28 | per day | estimated |

${\beta}_{As}$ | 0.01–0.2 | per day | estimated |

${\beta}_{Am}$ | 0.01–0.2 | per day | estimated |

$\delta $ | 0.2 | per day | estimated |

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## Share and Cite

**MDPI and ACS Style**

Murillo, D.; Murillo, A.; Lee, S.
The Role of Vertical Transmission in the Control of Dengue Fever. *Int. J. Environ. Res. Public Health* **2019**, *16*, 803.
https://doi.org/10.3390/ijerph16050803

**AMA Style**

Murillo D, Murillo A, Lee S.
The Role of Vertical Transmission in the Control of Dengue Fever. *International Journal of Environmental Research and Public Health*. 2019; 16(5):803.
https://doi.org/10.3390/ijerph16050803

**Chicago/Turabian Style**

Murillo, David, Anarina Murillo, and Sunmi Lee.
2019. "The Role of Vertical Transmission in the Control of Dengue Fever" *International Journal of Environmental Research and Public Health* 16, no. 5: 803.
https://doi.org/10.3390/ijerph16050803