# The Influence of Anthropogenic and Environmental Disturbances on Parameter Estimation of a Dengue Transmission Model

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Characteristics of the Study Area

#### 2.2. Experimental Data

#### 2.3. Mathematical Model and Control Actions

#### 2.4. Parameter Estimation

- A.
- We fit one epidemic outbreak over 93 epidemiological weeks, without pulse-type inputs (${u}_{m}=0$ in Equation (2)), i.e., we assume that there were no external dynamics that could affect the mosquitoes population (no chemical control or environmental changes);
- B.
- We fit one epidemic outbreak over 93 epidemiological weeks, but with the addition of one pulse-type input, which describes an external change that perturbs the mosquito population through a chemical control (${u}_{m}={u}_{{m}_{1}}$ in Equation (2)). Here, we estimated the parameters of the model and the pulse input together;
- C.
- We fit two epidemic outbreaks over 265 epidemiological weeks (covering 93 weeks of previous cases) and four pulse-type inputs: two positive inputs for two chemical control actions (${u}_{{m}_{1}}$, ${u}_{{m}_{4}}$) and two negative inputs (${u}_{{m}_{2}}$, ${u}_{{m}_{3}}$) for modeling an increase in mosquito mortality due to some favorable environmental conditions. In addition, in this case, we estimate simultaneously the four inputs and the model parameters.

#### 2.5. Confidence Sub-Contour Box

#### 2.6. Sensitivity and Uncertainty Analyses

## 3. Results

#### 3.1. Parameter Estimations with Zero, One, and Four Pulse-Type Inputs

Case A (Zero Pulse Input) | Case B (One Pulse Input) | Case C (Four Pulse Inputs) | ||||||
---|---|---|---|---|---|---|---|---|

Factor | Biological Interval | Estimation Interval | Nominal Value | CSB | Nominal Value | CSB | Nominal Value | CSB |

$E\left(0\right)$ | - | (0, 30,000) | 17,000 | (13,000, 23,000) | 21,600 | (18,800, 24,900) | 9610 | (9590, 12,800) |

$L\left(0\right)$ | - | (0, 30,000) | 3400 | (2400, 4300) | 13,200 | (11,000, 19,000) | 26,000 | (21,000, 35,000) |

$P\left(0\right)$ | - | (0, 30,000) | 5400 | (3900, 7500) | 12,000 | (9700, 16,000) | 21,900 | (20,900, 22,100) |

${M}_{s}\left(0\right)$ | - | (10,000, 10,000,000) | 4,800,000 | (4,400,000, 5,600,000) | 3,400,000 | (2,700,000, 3,800,000) | 8,100,000 | (7,700,000, 10,000,000) |

${M}_{e}\left(0\right)$ | - | (100, 1200) | 1000 | (980, 1200) | 310 | (244, 336) | 320 | (280, 330) |

${M}_{i}\left(0\right)$ | - | (0, 100) | 0.150000 | (0.110000, 0.210000) | 13 | (12, 16) | 16 | (15, 22) |

${H}_{s}\left(0\right)$ | (0, 400,000) | (0, 450,000) | 358,000 | (343,000, 414,000) | 160,000 | (150,000, 190,000) | 180,000 | (170,000, 200,000) |

${H}_{e}\left(0\right)$ | - | (0, 100) | 0.28 | (0.21, 0.41) | 10 | (9, 13) | 22 | (21, 25) |

$\delta $ | (65, 165) | (20, 180) | 92 | (64, 120) | 46 | (38, 59) | 49 | (42, 58) |

C | (6400, 95,000) | (6400, 340,000) | 120,000 | (95,000, 180,000) | 250,000 | (238,000, 290,000) | 231,000 | (199,000, 238,000) |

${\gamma}_{e}$ | (0.6, 2.3) | (0, 2.3) | 0.120 | (0.099, 0.170) | 1.29 | (1.10, 1.43) | 1.48 | (1.28, 1.52) |

${\mu}_{e}$ | - | (0, 1.3) | 0.0078 | (0.0056, 0.0101) | 0.90 | (0.84, 1.30) | 1.23 | (1.14, 1.25) |

${\gamma}_{l}$ | (0.05, 0.5) | (0, 1.6) | 0.42 | (0.32, 0.59) | 0.70 | (0.68, 0.88) | 0.86 | (0.83, 1.07) |

${\mu}_{l}$ | (0.07, 3.22) | (0, 3.22) | 2.7 | (2.0, 3.9) | 1.53 | (1.45, 1.88) | 1.43 | (1.36, 1.67) |

${\gamma}_{p}$ | (0.1, 1) | (0, 1.7) | 0.497 | (0.415, 0.696) | 0.91 | (0.75, 0.97) | 0.905 | (0.902, 1.110) |

${\mu}_{p}$ | (0, 1.4) | (0, 1.4) | 1.20 | (0.91, 1.75) | 0.54 | (0.44, 0.65) | 0.61 | (0.60, 0.66) |

f | (0.4, 0.6) | (0.3, 0.7) | 0.39 | (0.29, 0.52) | 0.49 | (0.42, 0.50) | 0.506 | (0.449, 0.522) |

${\beta}_{m}$ | (0, 4) | (0, 4) | 0.040 | (0.032, 0.052) | 1.52 | (1.50, 1.60) | 2.02 | (2.01, 2.03) |

${\mu}_{m}$ | (0.06, 0.3) | (0, 0.9) | 0.268 | (0.244, 0.270) | 0.449 | (0.449, 0.456) | 0.5360 | (0.5357, 0.5390) |

$\alpha $ | (1, 1.6) | (1, 1.6) | 1.03 | (1.03, 1.07) | 1.44 | (1.44, 1.46) | 1.4770 | (1.4668, 1.4773) |

${\theta}_{m}$ | (0.58, 0.88) | (0.4, 1.0) | 0.40 | (0.29, 0.51) | 0.634 | (0.630, 0.660) | 0.642 | (0.629, 0.643) |

${\mu}_{h}$ | - | (0.00001, 0.0009) | 0.000021 | (0.000016, 0.000029) | 0.000228 | (0.000190, 0.000302) | 0.000748 | (0.000587, 0.000788) |

${\beta}_{h}$ | (0, 4) | (0, 4) | 0.227 | (0.216, 0.249) | 1.43 | (1.39, 1.43) | 1.43 | (1.42, 1.44) |

${\theta}_{h}$ | (0.7, 1.75) | (0.4, 1.8) | 0.400 | (0.290, 0.430) | 0.70 | (0.61, 0.72) | 0.48 | (0.45, 0.49) |

${\gamma}_{h}$ | (0.5, 1.75) | (0.3, 2.0) | 0.328 | (0.322, 0.381) | 1.69 | (1.65, 1.69) | 1.65 | (1.65, 1.67) |

${A}_{{m}_{1}}$ | - | (0, 2) | - | - | 0.48 | (0.45, 0.57) | 0.69 | (0.62, 0.72) |

${t}_{{0}_{{c}_{1}}}$ | - | (32, 38) | - | - | 35.60 | (35.00, 36.00) | 35.80 | (35.60, 36.80) |

$\Delta {t}_{{c}_{1}}$ | (0, 12) | (0, 12) | - | - | 9.7 | (7.9, 11.5) | 11.99 | (11.39, 12.25) |

${A}_{{m}_{2}}$ | - | (−1.5, 0) | - | - | - | - | −0.46 | (−0.54, 0.45) |

${t}_{{0}_{{c}_{2}}}$ | - | (120, 134) | - | - | - | - | 132 | (125, 145) |

$\Delta {t}_{{c}_{2}}$ | - | (0, 12) | - | - | - | - | 6 | (5, 7) |

${A}_{{m}_{3}}$ | - | (−1.5, 0) | - | - | - | - | −0.712 | (−0.919, −0.709) |

${t}_{{0}_{{c}_{3}}}$ | - | (210, 235) | - | - | - | - | 231 | (227, 233) |

$\Delta {t}_{{c}_{3}}$ | - | (0, 12) | - | - | - | - | 5.67 | (4.86, 5.86) |

${A}_{{m}_{4}}$ | - | (0, 2) | - | - | - | - | 0.34 | (0.32, 0.36) |

${t}_{{0}_{{c}_{4}}}$ | - | (240, 260) | - | - | - | - | 243 | (240, 246) |

$\Delta {t}_{{c}_{4}}$ | (0, 12) | (0, 12) | - | - | - | - | 11 | (10, 12) |

#### 3.2. Estimation of a Sub-Contour Box for Nominal Parameters

#### 3.3. SA: Parameters That Determine the Model Behavior

#### 3.4. Simulation of Control Strategies

- Effect of positive and negative input amplitudes over the vector populations (aquatic and adult stages);
- Variation in pulse-type chemical control input parameters (${A}_{{c}_{j}}$ and ${t}_{{0}_{{c}_{j}}}$, with j = {1,4});
- Human immunization (vaccination) as a pulse-type input similar to chemical control (see Section 2.3).

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Reproductive Number R 0 and Equilibrium

## References

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**Figure 1.**Estimated outputs (${H}_{i}$) using the estimated nominal values in Table 2 for cases A, B, and C. The pulse-type input has the same units as the mortality rate (mosquitoes per week). (

**a**) Estimated output for one dengue outbreak without pulse-type input; (

**b**) estimated output for one dengue outbreak with one pulse-type input; (

**c**) estimated output for two dengue outbreaks with four pulse-type inputs.

**Figure 2.**Uncertainty analysis for the dengue model in cases A, B, and C using CSB intervals from Table 2 with 1000 simulations. The CSB method guarantees that at least 95% of curves generated from random combinations of parameters produce outputs classified as ‘right’ outputs. (

**a**) Uncertainty analysis for dengue model with non pulses input; (

**b**) uncertainty analysis for dengue model with one pulse input; (

**c**) uncertainty analysis for a dengue model with four pulse inputs.

**Figure 3.**Sensitivity analysis for the CSB intervals. (

**a**) The scalar (MSE) SA results for case C and graphical CSB validation using the Xiao method and CSB intervals; (

**b**) results of the vectorial SA plotted by time for case C and graphical CSB validation using Xiao’s method with N = 6000 [49].

**Figure 4.**Boxplot for all parameters estimated with the four pulse-type inputs model (Case C). The boxes and whiskers are filtered estimations that define the medians (nominal value), and the red stars represent the biological intervals identified in the literature. All values are normalized according to the minimum and maximum values of the estimation intervals proposed in Table 2.

**Figure 5.**Sensitivity analysis for the dengue model with four pulse-type inputs for case C (with 1% uncertainty in each nominal parameter), using (

**a**) scalar and (

**b**) vectorial $S{T}_{i}$ indices from the Saltelli method.

**Figure 6.**Simulation of case C focusing on the vector population. We used nominal values to observe how the mosquito and aquatic phases are affected by pulse-type inputs. The simulation began in week five because of the quick and significant decrease of the state variables in the vector population. The amplitude of the pulses in the plots is amplified by a factor of 10 for better visualization.

**Figure 7.**Monte Carlo simulations of the dengue transmission model for the number of infected humans in two scenarios: (

**a**) increasing the value of ${A}_{{c}_{1}}$ between 30–60% and 60–90% of its nominal value; (

**b**) increasing and decreasing the value of ${A}_{{C}_{4}}$ between 20% and 60% of its nominal value; (

**c**,

**d**) late and early chemical control, increasing or decreasing ${t}_{{0}_{{c}_{1}}}$ and ${t}_{{0}_{{c}_{4}}}$ between 20–60% and 10–20% of their nominal values, respectively. For each scenario, we performed 1000 simulations.

**Figure 8.**Dengue model simulations using estimated nominal values and immunization control by vaccination. We removed chemical control pulses for immunization simulations and left those that represented the mosquito population increase. (

**a**) immunization of the 10 to 20% of total human population; (

**b**) immunization of the 20 to 30% of total human population.

**Table 1.**Definition of state variables, parameters, and input variables for the mathematical model (1). The input that describes mosquitoes growth (${u}_{m}$) is defined by three parameters (${A}_{{c}_{j}}$, ${t}_{{0}_{{c}_{j}}}$, and $\Delta {t}_{{c}_{j}}$) in Equation (2); in addition, ${u}_{m}$ could be formed by multiple j pulse-type inputs as described in Section 2.4. Note that, for all factors, the time unit is one week, and the rate unit is [week

^{−1}].

Factors | Description | Factors | Description |
---|---|---|---|

E | Number of eggs | $\delta $ | Oviposition rate |

L | Number of larvae | C | Egg carrying capacity |

P | Number of pupae | ${\gamma}_{e}$ | Egg to larva transition rate |

${M}_{s}$ | Number of susceptible mosquitoes | ${\mu}_{e}$ | Egg mortality rate |

${M}_{e}$ | Number of exposed mosquitoes | ${\gamma}_{l}$ | Larva to pupa transition rate |

${M}_{i}$ | Number of infected mosquitoes | ${\mu}_{l}$ | Larvae mortality rate |

${H}_{s}$ | Number of susceptible humans | ${\gamma}_{p}$ | Pupae to mosquito transition rate |

${H}_{e}$ | Number of exposed humans | ${\mu}_{p}$ | Pupae mortality rate |

${H}_{i}$ | Number of infected humans | f | Fraction of females that emerges |

${H}_{r}$ | Number of recovered humans | ${\beta}_{m}$ | Transmission coefficient human-mosquito |

M | Total number of mosquitoes | ${\mu}_{m}$ | Mosquito mortality rate |

H | Total number of humans | $\alpha $ | Change in ${\mu}_{m}$ due to virus infection |

${u}_{e}$ | Egg control input rate | ${\theta}_{m}$ | Extrinsic incubation rate |

${u}_{l}$ | Larvae control input rate | ${\mu}_{h}$ | Human mortality rate |

${u}_{p}$ | Pupae control input rate | ${\beta}_{h}$ | Transmission coefficient mosquito-human |

${u}_{m}$ | Mosquito control input rate | ${\theta}_{h}$ | Intrinsic incubation rate |

${u}_{v}$ | Vaccine control input rate | ${\gamma}_{h}$ | Human recovery rate |

${A}_{{c}_{j}}$ | Mosquito control pulse amplitude | ||

${t}_{{0}_{{c}_{j}}}$ | Mosquito control initial time | ||

$\Delta {t}_{{c}_{j}}$ | Mosquito control pulse width |

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## Share and Cite

**MDPI and ACS Style**

Catano-Lopez, A.; Rojas-Diaz, D.; Vélez, C.M.
The Influence of Anthropogenic and Environmental Disturbances on Parameter Estimation of a Dengue Transmission Model. *Trop. Med. Infect. Dis.* **2023**, *8*, 5.
https://doi.org/10.3390/tropicalmed8010005

**AMA Style**

Catano-Lopez A, Rojas-Diaz D, Vélez CM.
The Influence of Anthropogenic and Environmental Disturbances on Parameter Estimation of a Dengue Transmission Model. *Tropical Medicine and Infectious Disease*. 2023; 8(1):5.
https://doi.org/10.3390/tropicalmed8010005

**Chicago/Turabian Style**

Catano-Lopez, Alexandra, Daniel Rojas-Diaz, and Carlos M. Vélez.
2023. "The Influence of Anthropogenic and Environmental Disturbances on Parameter Estimation of a Dengue Transmission Model" *Tropical Medicine and Infectious Disease* 8, no. 1: 5.
https://doi.org/10.3390/tropicalmed8010005