# Sex-Specific Asymmetrical Attack Rates in Combined Sexual-Vectorial Transmission Epidemics

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Attack Rates by Sex

#### 2.2. Sexual Force of Infection

#### 2.3. Epidemic Threshold and ${\mathcal{R}}_{0}$

#### 2.4. Fitting the Model to Data

## 3. Results

#### 3.1. Sexual Force of Infection

#### 3.2. The Basic Reproduction Number: ${\mathcal{R}}_{0}$

#### 3.3. Fitting the Model to Data

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Reiner, R.C., Jr.; Perkins, T.A.; Barker, C.M.; Niu, T.; Chaves, L.F.; Ellis, A.M.; George, D.B.; Le Menach, A.; Pulliam, J.R.; Bisanzio, D.; et al. A systematic review of mathematical models of mosquito-borne pathogen transmission: 1970–2010. J. R. Soc. Interface
**2013**, 10, 20120921. [Google Scholar] [CrossRef] [PubMed] - Anderson, R.M.; Garnett, G.P. Mathematical models of the transmission and control of sexually transmitted diseases. Sex. Transmitted Dis.
**2000**, 27, 636–643. [Google Scholar] [CrossRef] - Smith, D.L.; Perkins, T.A.; Reiner, R.C., Jr.; Barker, C.M.; Niu, T.; Chaves, L.F.; Ellis, A.M.; George, D.B.; Le Menach, A.; Pulliam, J.R.; et al. Recasting the theory of mosquito-borne pathogen transmission dynamics and control. Trans. R. Soc. Trop. Med. Hyg.
**2014**, 108, 185–197. [Google Scholar] [CrossRef][Green Version] - Garnett, G.P. An introduction to mathematical models in sexually transmitted disease epidemiology. Sex. Transmitted Infect.
**2002**, 78, 7–12. [Google Scholar] [CrossRef][Green Version] - Lucey, D.R.; Gostin, L.O. The Emerging Zika Pandemic. JAMA
**2016**, 315, 865. [Google Scholar] [CrossRef] - Nunes, M.L.; Carlini, C.R.; Marinowic, D.; Neto, F.K.; Fiori, H.H.; Scotta, M.C.; Zanella, P.L.Á.; Soder, R.B.; da Costa, J.C. Microcephaly and Zika virus: A clinical and epidemiological analysis of the current outbreak in Brazil. J. Pediatr. (Versão em Português)
**2016**, 92, 230–240. [Google Scholar] [CrossRef] - Wikan, N.; Smith, D.R. Zika virus: History of a newly emerging arbovirus. Lancet Infect. Dis.
**2016**, 16, e119–e126. [Google Scholar] [CrossRef] - Abushouk, A.I.; Negida, A.; Ahmed, H. An updated review of Zika virus. J. Clin. Virol.
**2016**, 84, 53–58. [Google Scholar] [CrossRef] [PubMed] - Zanluca, C.; de Melo, V.C.A.; Mosimann, A.L.P.; dos Santos, G.I.V.; dos Santos, C.N.D.; Luz, K. First report of autochthonous transmission of Zika virus in Brazil. Memórias do Instituto Oswaldo Cruz
**2015**, 110, 569–572. [Google Scholar] [CrossRef] [PubMed][Green Version] - Heukelbach, J.; Alencar, C.H.; Kelvin, A.A.; De Oliveira, W.K.; Pamplona de Góes Cavalcanti, L. Zika virus outbreak in Brazil. J. Infect. Dev. Ctries.
**2016**, 10, 116–120. [Google Scholar] [CrossRef] - Paixão, E.S.; Barreto, F.; da Glória Teixeira, M.; da Conceição, N.; Costa, M.; Rodrigues, L.C. History, Epidemiology, and Clinical Manifestations of Zika: A Systematic Review. Am. J. Public Health
**2016**, 106, 606–612. [Google Scholar] [CrossRef][Green Version] - Atkinson, B.; Hearn, P.; Afrough, B.; Lumley, S.; Carter, D.; Aarons, E.J.; Simpson, A.J.; Brooks, T.J.; Hewson, R. Detection of zika virus in semen. Emerg. Infect. Dis.
**2016**, 22, 940. [Google Scholar] [CrossRef] [PubMed] - Mansuy, J.M.; Dutertre, M.; Mengelle, C.; Fourcade, C.; Marchou, B.; Delobel, P.; Izopet, J.; Martin-Blondel, G. Zika virus: High infectious viral load in semen, a new sexually transmitted pathogen? Lancet Infect. Dis.
**2016**, 16, 405. [Google Scholar] [CrossRef] - Coelho, F.C.; Durovni, B.; Saraceni, V.; Lemos, C.; Codeço, C.T.; Camargo, S.; De Carvalho, L.M.; Bastos, L.; Arduini, D.; Villela, D.A.; et al. Higher incidence of Zika in adult women than adult men in Rio de Janeiro suggests a significant contribution of sexual transmission from men to women. Int. J. Infect. Dis.
**2016**, 51, 128–132. [Google Scholar] [CrossRef] [PubMed][Green Version] - Davidson, A.; Slavinski, S.; Komoto, K.; Rakeman, J.; Weiss, D. Suspected Female-to-Male Sexual Transmission of Zika Virus—New York City, 2016. MMWR Morb. Mortal. Wkly. Rep.
**2016**, 65, 716–717. [Google Scholar] [CrossRef] [PubMed] - Deckard, D.T.; Chung, W.M.; Brooks, J.T.; Smith, J.C.; Woldai, S.; Hennessey, M.; Kwit, N.; Mead, P. Male-to-Male Sexual Transmission of Zika Virus—Texas, January 2016. MMWR Morb. Mortal. Wkly. Rep.
**2016**, 65, 372–374. [Google Scholar] [CrossRef] [PubMed] - Towers, S.; Brauer, F.; Castillo-Chavez, C.; Falconar, A.K.; Mubayi, A.; Romero-Vivas, C.M. Estimate of the reproduction number of the 2015 Zika virus outbreak in Barranquilla, Colombia, and estimation of the relative role of sexual transmission. Epidemics
**2016**, 17, 50–55. [Google Scholar] [CrossRef][Green Version] - Gao, D.; Lou, Y.; He, D.; Porco, T.C.; Kuang, Y.; Chowell, G.; Ruan, S. Prevention and control of Zika fever as a mosquito-borne and sexually transmitted disease: A Mathematical Modeling Analysis. Sci. Rep.
**2016**, 6, 28070. [Google Scholar] [CrossRef] [PubMed] - Baca-Carrasco, D.; Velasco-Hernández, J.X. Sex, Mosquitoes and Epidemics: An Evaluation of Zika Disease Dynamics. Bull. Math. Biol.
**2016**, 78, 2228–2242. [Google Scholar] [CrossRef] - Isea, R.; Lonngren, K.E. A Preliminary Mathematical Model for the Dynamic Transmission of Dengue, Chikungunya and Zika. Am. J. Mod. Phys. Appl.
**2016**, 3, 11–15. [Google Scholar] - Bastos, M.M.; Coelho, F.C. Estimating under-observation and the full size of the 2016 Zika epidemic in Rio de Janeiro. PLoS ONE
**2018**, 13, e0205001. [Google Scholar] [CrossRef] [PubMed] - Coelho, F.C.; Codeço, C.T.; Gomes, M.G.M. A bayesian framework for parameter estimation in dynamical models. PLoS ONE
**2011**, 6, e19616. [Google Scholar] [CrossRef] [PubMed] - Villela, D.; Bastos, L.; De Carvalho, L.; Cruz, O.; Gomes, M.; Durovni, B.; Lemos, M.; Saraceni, V.; Coelho, F.; Codeço, C. Zika in Rio de Janeiro: Assessment of basic reproduction number and comparison with dengue outbreaks. Epidemiol. Infect.
**2017**, 145, 1649–1657. [Google Scholar] [CrossRef] [PubMed] - Van den Driessche, P.; Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci.
**2002**, 180, 29–48. [Google Scholar] [CrossRef] - Anderson, R.M.; May, R.M. Infectious Diseases of Humans: Dynamics and Control; Oxford University Press: Oxford, UK, 1992. [Google Scholar]
- Santos, B.M.C.d.; Coelho, F.C.; Armstrong, M.; Saraceni, V.; Lemos, C. Zika: An ongoing threat to women and infants. Cad. Saude Publica
**2018**, 34, e00038218. [Google Scholar] [CrossRef] [PubMed] - Pacheco, O.; Beltrán, M.; Nelson, C.A.; Valencia, D.; Tolosa, N.; Farr, S.L.; Padilla, A.V.; Tong, V.T.; Cuevas, E.L.; Espinosa-Bode, A.; et al. Zika virus disease in Colombia—Preliminary report. N. Engl. J. Med.
**2016**. [Google Scholar] [CrossRef] - Hess, K.L.; Crepaz, N.; Rose, C.; Purcell, D.; Paz-Bailey, G. Trends in sexual behavior among men who have sex with men (MSM) in high-income countries, 1990–2013: A systematic review. AIDS Behav.
**2017**, 21, 2811–2834. [Google Scholar] [CrossRef] [PubMed] - Atkinson, B.; Thorburn, F.; Petridou, C.; Bailey, D.; Hewson, R.; Simpson, A.J.; Brooks, T.J.; Aarons, E.J. Presence and persistence of Zika virus RNA in semen, United Kingdom, 2016. Emerg. Infect. Dis.
**2017**, 23, 611. [Google Scholar] [CrossRef] [PubMed] - Kim, C.R.; Counotte, M.; Bernstein, K.; Deal, C.; Mayaud, P.; Low, N.; Broutet, N. Investigating the sexual transmission of Zika virus. Lancet Glob. Health
**2018**, 6, e24–e25. [Google Scholar] [CrossRef] - Folkers, K.M.; Caplan, A.L.; Igel, L.H. Zika, sexual transmission and prudent public health policy. Public Health
**2017**, 148, 66–68. [Google Scholar] [CrossRef]

**Figure 1.**Simulation of the model’s dynamics with ${\beta}_{s}=0.25$, ${\beta}_{v}=0.01$, $\mu =0.1$, $e=0.2$, ${\tau}_{l}=0.01$, $\rho =1$, and ${K}_{L}=1$. ${\mathcal{R}}_{0}=3.04$, which is compatible with values reported by Villela et al. [23]. The $A{R}_{W}$ and $A{R}_{W}$ curves correspond to the attack rates over time for women and men, respectively.

**Figure 2.**Sexual force of infection for women with the same parameters as those of Figure 1. In the top panel, we can observe that the sexual force of infection of women (${\lambda}_{SW}\left(t\right)$) remains elevated for quite a longer period of time if compared to the vectorial force of infection (${\lambda}_{V}\left(t\right)$) shown in the lower panel.

**Figure 3.**Sexual force of infection as a function of ${k}_{L}$, or how effective the sexual transmission from men to women is in the post-viremic phase. Notice that in the absence of effective longer term sexual transmission from men to women, the dynamics reverts to that of a standard vector-borne infection.

**Figure 4.**Qualitative differences between the impact of sexual bias in reporting, namely underreporting of male cases (

**left**panel) and sexual transmission in the prevalence curves ${W}_{I}\left(t\right)$ and ${M}_{I}\left(t\right)$ (

**right**panel). Notice that the crossing of the prevalence curves indicates the presence of sexual transmission as this can never happen from underreporting alone.

**Figure 5.**Ratio $\frac{A{R}_{W}\left(120\right)}{A{R}_{M}\left(120\right)}$ for a range of ${\beta}_{s}$ and ${\beta}_{v}$ values. The green line represents ${\mathcal{R}}_{0}=1$, i.e., the epidemic threshold. Any point to the right of this curve has ${\mathcal{R}}_{0}>1$. It is worth noticing that the reported excess cases reported for Zika in women are possible both during epidemics and off-season [26].

**Figure 6.**Joint posterior distribution of the transmission parameters ${\beta}_{s}$ and ${\beta}_{v}$.

**Figure 7.**${\mathcal{R}}_{0}$ as a function of the relative intensities of sexual (${\beta}_{s}$) and vectorial (${\beta}_{v}$) transmissions. The ${\mathcal{R}}_{0}$ values are already adjusted for the heterogeneity in sexual contact rates.

**Figure 9.**Posterior distributions of ${W}_{I}\left(t\right)$ (top panel) and ${M}_{I}\left(t\right)$ (bottom panel). Shaded areas represent 95% credibility intervals. Blue dots are the data. The Y-axis is the prevalence as a fraction of the population.

**Table 1.**Variables and parameters of the model. Values obtained from the literature are marked with references numbers. Ranges marked with a ${}^{\u2020}$ correspond to values explored in simulations, but for which no experimental data could be found. * Fraction of the entire population, N.

Symbols | Description | Value Range |
---|---|---|

${W}_{S}$ | Susceptible women | $[0,1]$ * |

${W}_{E}$ | Exposed women | $[0,1]$ * |

${W}_{I}$ | Infectious women | $[0,1]$ * |

${M}_{S}$ | Susceptible men | $[0,1]$ * |

${M}_{E}$ | Exposed men | $[0,1]$ * |

${M}_{I}$ | Infectious men | $[0,1]$ * |

${M}_{L}$ | Latent men | $[0,1]$ * |

$\mu $ | Recovery rate (day${}^{-1}$) | $[0.001,0.1]$ [18] |

${\beta}_{V}$ | Vector transmission rate (day${}^{-1}$) | $[0.1,0.75]$ [18] |

${\beta}_{S}$ | Sexual transmission rate ((people × day)${}^{-1}$) | $[0,2]{\phantom{\rule{3.33333pt}{0ex}}}^{\u2020}$ |

${k}_{WW}$ | Women-to-women transmissibility modifier | $[0,1]{\phantom{\rule{3.33333pt}{0ex}}}^{\u2020}$ |

${k}_{WM}$ | Women-to-men transmissibility modifier | $[0,1]{\phantom{\rule{3.33333pt}{0ex}}}^{\u2020}$ |

${k}_{MM}$ | Men-to-men transmissibility modifier | $[0,1]{\phantom{\rule{3.33333pt}{0ex}}}^{\u2020}$ |

${k}_{L}$ | Latent period transmissibility modifier | $[0,1]{\phantom{\rule{3.33333pt}{0ex}}}^{\u2020}$ |

e | Incubation rate (day${}^{-1}$) | $[0.14,0.5]$ [18] |

$\rho $ | Fraction of men becoming latent | $[0,1]{\phantom{\rule{3.33333pt}{0ex}}}^{\u2020}$ |

${\tau}_{l}$ | Latent recovery rate (day${}^{-1}$) | $[0.025,0.1]$ [12] |

**Table 2.**Parameters estimated from data, along with prior and posterior distributions. ${}^{\u2020}$ The exponential distribution is parameterized in standardized form (loc, scale). Posteriors are given as medians and the 95% credible interval. All remaining parameters were kept constant.

Parameter | Prior | Posterior |
---|---|---|

${\beta}_{s}$ | $\mathcal{N}(0.98,0.3)$ | $0.99[0.02,1.54]$ |

${\beta}_{v}$ | $\mathcal{N}(0.25,0.1)$ | $0.33[0.24,0.61]$ |

$\mu $ | $\mathcal{N}(0.15,0.1)$ | $0.17[0.10,0.23]$ |

${\tau}_{l}$ | $Exp(0.0001,0.1){\phantom{\rule{3.33333pt}{0ex}}}^{\u2020}$ | $0.7[0.005,0.37]$ |

$ur$ | $\mathcal{U}(0,1)$ | $0.81[0.44,0.99]$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

de Barros, A.C.W.G.; Santos, K.G.; Massad, E.; Coelho, F.C.
Sex-Specific Asymmetrical Attack Rates in Combined Sexual-Vectorial Transmission Epidemics. *Microorganisms* **2019**, *7*, 112.
https://doi.org/10.3390/microorganisms7040112

**AMA Style**

de Barros ACWG, Santos KG, Massad E, Coelho FC.
Sex-Specific Asymmetrical Attack Rates in Combined Sexual-Vectorial Transmission Epidemics. *Microorganisms*. 2019; 7(4):112.
https://doi.org/10.3390/microorganisms7040112

**Chicago/Turabian Style**

de Barros, Ana Carolina W. G., Kaline G. Santos, Eduardo Massad, and Flávio Codeço Coelho.
2019. "Sex-Specific Asymmetrical Attack Rates in Combined Sexual-Vectorial Transmission Epidemics" *Microorganisms* 7, no. 4: 112.
https://doi.org/10.3390/microorganisms7040112