# Mathematical Modeling and Characterization of the Spread of Chikungunya in Colombia

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

- The total population of humans ${N}_{h}(t)$ is divided into five subpopulations: humans who may become infected (susceptible ${S}_{h}(t)$), humans exposed, but still not infected due to the existence of an incubation period of the virus (latent ${E}_{h}(t)$), humans infected by the Chikungunya virus and that develop the disease (infected ${I}_{h}(t)$), humans who have recovered from the Chikungunya infection (recovered ${R}_{h}(t)$), and humans who have the disease chronically (chronic ${C}_{h}(t)$).
- The parameter ${\mu}_{h}$ is the birth rate of humans. The birth rate ${\mu}_{h}$ is assumed equal to natural death ${d}_{h}$.
- The mortality rate increase due to the disease is a real fact. However, since this rate is small in comparison with other rates and is not going to affect the dynamics, we assume that $\u03f5=0$.
- The total population of mosquitoes ${N}_{v}(t)$ is divided into three subpopulations: mosquitoes who may become infected (susceptible ${S}_{v}(t)$), mosquitoes in a latent stage (latent ${E}_{v}(t)$), and mosquitoes currently infected or spreading the Chikungunya virus (infected ${I}_{v}(t)$).
- The parameter ${\mu}_{v}$ is the birth rate of the mosquitoes, and it is assumed equal to the death rate ${d}_{v}$.
- A susceptible human can transit to the latent subpopulation ${E}_{h}(t)$ because of an effective transmission due to a bite of an infected mosquito at a rate of ${\beta}_{1}^{\prime}$.
- A susceptible mosquito can be infected if there exists an effective transmission when it bites an infected human, at a rate ${\beta}_{2}$.
- A fraction $\alpha $ of the latent humans passes to infected by the virus.
- A fraction $\gamma $ of the infected humans recovers, i.e., they do not have the disease anymore.
- A fraction $\rho $ of the recovered humans moves to the chronic class.
- A fraction $\varphi $ of the latent mosquitoes goes through to infected mosquitoes.
- Homogeneous mixing is assumed, i.e., all susceptible humans have the same probability of being infected and all susceptible mosquitoes have the same probability of being infected.

## 3. Analysis of the Model

#### 3.1. Equilibrium Points and Local Stability of the Chikungunya Mathematical Model

**Theorem**

**1.**

#### 3.2. Endemic Equilibria

#### 3.3. Global Stability Analysis

**Theorem**

**2.**

**Proof.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Numerical Simulation

#### 4.1. Numerical Simulations for ${\mathcal{R}}_{0}<1$

^{−1}, except ${\beta}_{i}$, which measures the effective contacts per day, i.e., the total number of contacts, effective or not, per unit day, multiplied by the risk of infection with Chikungunya virus.

#### 4.2. Numerical Simulations for ${\mathcal{R}}_{0}>1$

#### 4.3. Sensitivity Analysis of the Transmission Parameters

## 5. Estimation of Parameters for the Colombian Scenario

#### 5.1. Fitting Algorithm

- For a given $({\beta}_{1},{\beta}_{2},{i}_{v}(0))$, numerically solve the system of differential Equation (2) and obtain a solution ${\widehat{Y}}_{j}(t)=(\widehat{{s}_{hj}},\widehat{{e}_{hj}},\widehat{{i}_{hj}},\widehat{{r}_{hi}},\widehat{{c}_{hj}},\widehat{{s}_{vj}},\widehat{{e}_{vj}}$, $\widehat{{i}_{vj}}),$ which is an approximation of the real data solution $Y(t)$.
- Set ${t}_{0}=0$ (the fitting process starts at Week 0), and for $t=0,1,2,\dots ,51$, corresponding to weeks where data are available, evaluate the computed numerical solution for subpopulation ${i}_{h}(t)$; i.e., $\widehat{{i}_{h}}(0)$, $\widehat{{i}_{h}}(1)$, $\widehat{{i}_{h}}(2)$, …, $\widehat{{i}_{h}}(51)$.
- Compute the sum of square of the difference between $\widehat{{i}_{h}}(0)$, $\widehat{{i}_{h}}(1)$, $\widehat{{i}_{h}}(2)$, …, $\widehat{{i}_{h}}(51)$, and infectious data in Table 1. This function $\mathbb{F}$ returns the sum of squared errors ($SSR$), where for the Colombia data are given by:$$SSR=\sum _{j=0}^{51}{(\widehat{{i}_{h}}(j)-{i}_{h}(j))}^{2}$$
- Find a global minimum for the the sum of squared errors ($SSR$) using genetic, trust-region-reflective, and interior point algorithms.

#### 5.2. Numerical Simulation of the Chikungunya Mathematical Model

#### 5.3. Identifiability of the Parameters

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Dynamics of the Chikungunya virus with transmission vector. The boxes represent the subpopulation and the arrows the transition between the subpopulations. Arrows are labeled by their corresponding model parameters.

**Figure 4.**The dynamics of the different subpopulations are sensitive to the changes of the parameter ${\beta}_{2}$.

**Figure 5.**Best fit of the Chikungunya mathematical model (2) to the time-series data of Chikungunya in Colombia corresponding to the year 2015. Red points give the real data, and the blue line shows the best model fit. The best fit parameter values are given in the table. SSR, sum of squared errors.

Parameter | ${\beta}_{1}$ | ${\beta}_{2}$ | ${i}_{v}(0)$ | SSR | ${\mathcal{R}}_{0}$ |

Value | $0.361$ | 0.012 | 2.186 × 10^{−4} | 4.701 × 10^{−7} | 0.77 |

**Figure 6.**Dynamic of the chronic infected individual humans using the fit of the Chikungunya mathematical model (2) to the time-series data of Chikungunya in Colombia corresponding to the year 2015.

**Figure 7.**Best fits of the Chikungunya mathematical model (2) to the time-series data of Chikungunya in Colombia corresponding to the year 2015. The initial proportion of infected vectors ${i}_{v}(0)$ varies from 0.2 (20%) to 0.5 (50%).

**Figure 8.**Identifiability assessment of the mathematical model (2) fit to the prevalence data of Chikungunya in the population of Colombia in the year 2016. Correlation plots generated with parameter estimates from bootstrap fits.

**Figure 9.**Identifiability assessment of the mathematical model (2) fit to the prevalence data of Chikungunya in the population of Colombia in the year 2016. Correlation plots generated with parameter estimates from bootstrap fits. Correlation between the parameters ${\beta}_{1}$, ${\beta}_{2}$, ${i}_{M}(0)$, and ${\mathcal{R}}_{0}.$

**Figure 10.**Identifiability assessment of the mathematical model (2) fit to the prevalence data of Chikungunya in the population of Colombia in the year 2016. Correlation plots generated with Markov chain Monte Carlo for three (${\beta}_{1},{\beta}_{2}$, ${i}_{v}(0)$) and four parameters (${\beta}_{1},{\beta}_{2}$, ${i}_{v}(0)$, ${i}_{h}(0)$), respectively.

**Table 1.**Parameter values for mathematical model of Chikungunya (2).

Parameter | Symbol | Values | Rate |
---|---|---|---|

Average life-span of the human host [50] | $\frac{1}{{\mu}_{h}}$ | 25.000 Days | $0.00004$ |

Average life-span of the vector [51] | $\frac{1}{{\mu}_{v}}$ | 14 Days | $0.071\overline{33}$ |

Average incubation time of the virus (latency in the humans) [52] | $\frac{1}{\alpha}$ | 5–12 Days | $0.1\overline{33}$ |

Average incubation time of the virus (latency in the vector) [28,53] | $\frac{1}{\varphi}$ | 3 Days | $0.\overline{33}$ |

Average infection time (infection in humans) [51] | $\frac{1}{\gamma}$ | 5–15 Days | $0.066$ |

Chronic time [54,55] | $\frac{1}{\rho}$ | 300 Days | $0.00\overline{33}$ |

Value ${\mathit{\beta}}_{2}$ | $\frac{3}{8}$ | $\frac{14}{50}$ | $\frac{2}{25}$ |
---|---|---|---|

Value of ${\mathcal{R}}_{0}$ | $4.94$ | $4.27$ | $2.28$ |

Value of Infected Population | $7.142$ million | $6.993$ million | $5.551$ million |

**Table 3.**Data provided by the National Institute of Health-SIVIGILA, Colombia. The first row corresponds to the number of the week of the year 2015. The second row shows the number of infectious individuals for whom Chikungunya was detected in each week because those people went to see a doctor and it was reported.

Week | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Cases | 15,000 | 16,200 | 16,100 | 14,800 | 13,900 | 14,000 | 12,950 | 12,100 |

Week | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

Cases | 13,900 | 13,000 | 12,200 | 11,900 | 8300 | 12,200 | 12,000 | 11,800 |

Week | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

Cases | 13,700 | 13,600 | 11,900 | 11,700 | 11,500 | 9800 | 7900 | 8000 |

Week | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 |

Cases | 7800 | 6100 | 6200 | 5000 | 5700 | 4000 | 2900 | 2300 |

Week | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |

Cases | 2100 | 2000 | 1900 | 1900 | 1950 | 1800 | 1900 | 1700 |

Week | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 |

Cases | 1750 | 1750 | 1650 | 1600 | 1650 | 1500 | 1600 | 1000 |

Week | 49 | 50 | 51 | 52 | ||||

Cases | 1000 | 1600 | 1000 | 900 |

**Table 4.**Estimated parameters using the mathematical model (2) and the prevalence data of Chikungunya in the population of Colombia in the year 2016.

Parameters | ${\mathcal{R}}_{0}$ | ${\mathit{i}}_{\mathit{v}}(0)$ |
---|---|---|

Values | 0.78 (0.74–0.84) | 2.18 ${\times \phantom{\rule{3.33333pt}{0ex}}10}^{-4}$ (1.95 ${\times \phantom{\rule{3.33333pt}{0ex}}10}^{-4}$–2.45 ${\times \phantom{\rule{3.33333pt}{0ex}}10}^{-4})$ |

Parameters | ${\beta}_{1}$ | ${\beta}_{2}$ |

Values | 0.36 (0.34–0.38) | 0.012 (0.011–0.13) |

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**MDPI and ACS Style**

González-Parra, G.C.; Aranda, D.F.; Chen-Charpentier, B.; Díaz-Rodríguez, M.; Castellanos, J.E. Mathematical Modeling and Characterization of the Spread of Chikungunya in Colombia. *Math. Comput. Appl.* **2019**, *24*, 6.
https://doi.org/10.3390/mca24010006

**AMA Style**

González-Parra GC, Aranda DF, Chen-Charpentier B, Díaz-Rodríguez M, Castellanos JE. Mathematical Modeling and Characterization of the Spread of Chikungunya in Colombia. *Mathematical and Computational Applications*. 2019; 24(1):6.
https://doi.org/10.3390/mca24010006

**Chicago/Turabian Style**

González-Parra, Gilberto C., Diego F. Aranda, Benito Chen-Charpentier, Miguel Díaz-Rodríguez, and Jaime E. Castellanos. 2019. "Mathematical Modeling and Characterization of the Spread of Chikungunya in Colombia" *Mathematical and Computational Applications* 24, no. 1: 6.
https://doi.org/10.3390/mca24010006