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February  2018, 15(1): 125-140. doi: 10.3934/mbe.2018005

Sex-biased prevalence in infections with heterosexual, direct, and vector-mediated transmission: a theoretical analysis

1. 

Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14,38123 Povo (TN), Italy

2. 

School of Mathematical & Statistical Sciences, Arizona State University, Tempe, AZ 85281, USA

3. 

Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Ciudad Universitaria 04510, Ciudad de Mexico, Mexico

* Corresponding author

Received  October 30, 2016 Accepted  January 20, 2017 Published  May 2017

Three deterministic Kermack-McKendrick-type models for studying the transmission dynamics of an infection in a two-sex closed population are analyzed here. In each model it is assumed that infection can be transmitted through heterosexual contacts, and that there is a higher probability of transmission from one sex to the other than vice versa. The study is focused on understanding whether and how this bias in transmission reflects in sex differences in final attack ratios (i.e. the fraction of individuals of each sex that eventually gets infected). In the first model, where the other two transmission modes are not considered, the attack ratios (fractions of the population of each sex that will eventually be infected) can be obtained as solutions of a system of two nonlinear equations, that has a unique solution if the net reproduction number exceeds unity. It is also shown that the ratio of attack ratios depends solely on the ratio of gender-specific susceptibilities and on the basic reproductive number of the epidemic $ \mathcal{R}_0 $, and that the gender-specific final attack-ratio is biased in the same direction as the gender-specific susceptibilities. The second model allows also for infection transmission through direct, non-sexual, contacts. In this case too, an analytical expression is derived from which the attack ratios can be obtained. The qualitative results are similar to those obtained for the previous model, but another important parameter for determining the value of the ratio between the attack ratios in the two sexes is obtained, the relative weight of direct vs. heterosexual transmission (namely, ρ). Quantitatively, the ratio of final attack ratios generally will not exceed 1.5, if non-sexual transmission accounts for most transmission events (ρ ≥ 0.6) and the ratio of gender-specific susceptibilities is not too large (say, 5 at most).

The third model considers vector-borne, instead of direct transmission. In this case, we were not able to find an analytical expression for the final attack ratios, but used instead numerical simulations. The results on final attack ratios are actually quite similar to those obtained with the second model. It is interesting to note that transient patterns can differ from final attack ratios, as new cases will tend to occur more often in the more susceptible sex, while later depletion of susceptibles may bias the ratio in the opposite direction.

The analysis of these simple models, despite their lack of realism, can help in providing insight into, and assessment of, the potential role of gender-specific transmission in infections with multiple modes of transmission, such as Zika virus (ZIKV), by gauging what can be expected to be seen from epidemiological reports of new cases, disease incidence and seroprevalence surveys.

Citation: Andrea Pugliese, Abba B. Gumel, Fabio A. Milner, Jorge X. Velasco-Hernandez. Sex-biased prevalence in infections with heterosexual, direct, and vector-mediated transmission: a theoretical analysis. Mathematical Biosciences & Engineering, 2018, 15 (1) : 125-140. doi: 10.3934/mbe.2018005
References:
[1]

C. L. Althaus and N. Low, How Relevant Is Sexual Transmission of Zika Virus?, PLOS Med., 13 (2016), 1-3. doi: 10.1371/journal.pmed.1002157. Google Scholar

[2]

M.-C. BoilyR. F. BaggaleyL. WangB. MasseR. G. WhiteR. J. Hayes and M. Alary, Heterosexual risk of HIV-1 infection per sexual act: Systematic review and meta-analysis of observational studies, Lancet Infect. Dis., 9 (2009), 118-129. doi: 10.1016/S1473-3099(09)70021-0. Google Scholar

[3]

R. B. BrooksM. P. CarlosR. A. MyersM. Grace WhiteT. Bobo-LenociD. AplanD. Blythe and K. A. Feldman, Likely Sexual Transmission of Zika Virus from a Man with No Symptoms of Infection --Maryland, 2016, MMWR Morb Mortal Wkly Rep, 65 (2016), 915-916. doi: 10.15585/mmwr.mm6534e2. Google Scholar

[4]

CDC, Hepatitis A Questions and Answers for the Public, URL https://www.cdc.gov/hepatitis/hav/afaq.htm.Google Scholar

[5]

F. C. CoelhoB. DurovniV. SaraceniC. LemosC. T. CodecoS. CamargoL. M. de CarvalhoL. BastosD. ArduiniD. A. M. Villela and M. Armstrong, 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., 51 (2016), 128-132. doi: 10.1016/j.ijid.2016.08.023. Google Scholar

[6]

W. D. DavidsonS SlavinskiK. Komoto and J. Rakeman, Suspected Female-to-Male Sexual Transmission of Zika Virus-New York City, 2016, MMWR Morb Mortal Wkly Rep, 65 (2016), 716-717. Google Scholar

[7]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, New York: John Wiley and Sons, 2000. Google Scholar

[8]

M. R. DuffyT.-H. ChenW. T. HancockA. M. PowersJ. L. KoolR. S. LanciottiM. PretrickM. MarfelS. HolzbauerC. DubrayL. GuillaumotA. GriggsM. BelA. J. LambertJ. LavenO. KosoyA. PanellaB. J. BiggerstaffM. Fischer and E. B. Hayes, Zika virus outbreak on Yap Island, Federated States of Micronesia., N. Engl. J. Med., 360 (2009), 2536-2543. Google Scholar

[9]

N. M. Ferguson, Z. M. Cucunubá, I. Dorigatti, G. L. Nedjati-Gilani, C. A. Donnelly, M.- G. Basáñez, P. Nouvellet and J. Lessler, Countering the Zika epidemic in Latin America, Science (80-. )., 353 (2016), 353-354, URL http://www.sciencemag.org/cgi/content/full/science.aag0219/DC1.Google Scholar

[10]

K. Fonseca, B. Meatherall, D. Zarra, M. Drebot, J. MacDonald, K. Pabbaraju, S. Wong, P.Webster, R. Lindsay and R. Tellier, First case of zika virus infection in a returning canadian traveler, Am. J. Trop. Med. Hyg., 91 (2014), 1035-1038, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4228871/.Google Scholar

[11] P. J. Fos, Epidemiology Foundations: The Science of Public Health, Jossey-Bass, San Francisco, 2011. Google Scholar
[12]

E. B. Hayes, Zika Virus Outside Africa, Emerg. Infect. Dis., 15 (2009), 1347-1350, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2819875/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2819875/.Google Scholar

[13]

W. O. Kermack and A. G. McKendrick, A contributions to the mathematical theory of epidemics, Proc. R. Soc. London A, 115 (1927), 700-721. Google Scholar

[14]

A. J. KucharskiS. FunkR. M. EggoH.-P. MalletW. J. Edmunds and E. J. Nilles, Transmission dynamics of zika virus in island populations: A modelling analysis of the 2013-14 french polynesia outbreak, PLoS Negl. Trop. Dis, 10 (2016), e0004726. doi: 10.1371/journal.pntd.0004726. Google Scholar

[15]

W. D. Z. Lopes, J. D. Rodriguez, F. A. Souza, T. R. dos Santos, R. S. dos Santos, W. M. Rosanese, W. R. Z. Lopes, C. A. Sakamoto and A. J. da Costa, Sexual transmission of Toxoplasma gondii in sheep, Vet. Parasitol., 195 (2013), 47-56, URL http://linkinghub.elsevier.com/retrieve/pii/S0304401713000083.Google Scholar

[16]

G. MacDonald, The analysis of equilibrium in malaria, Trop Dis Bull, 49 (1952), 813-829. Google Scholar

[17]

J. M. MansuyC. PasquierM. DaudinS. Chapuy-RegaudN. MoinardC. ChevreauJ. IzopetC. Mengelle and L. Bujan, Zika virus in semen of a patient returning from a non-epidemic area, Lancet Infect. Dis., 16 (2016), 894-895. doi: 10.1016/S1473-3099(16)30153-0. Google Scholar

[18]

J. C. Miller, Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes, Infect. Dis. Model., 2 (2017), 35-55, URL http://linkinghub.elsevier.com/retrieve/pii/S2468042716300203.Google Scholar

[19]

R. Pellissier and A. Rousselot, Enquete serologique sur l'incidence des virus neurol ropes chez quelques singes de l'Afrique Equatoriale Francaise (French) [A Serological Investigation of the Incidence of Neurotropic Viruses in Certain Monkeys of French Equatorial Africa], Bull. Société Pathol. Exot., 47 (1954), 228-231. Google Scholar

[20]

A. J. Rodriguez-Morales, A. C. Bandeira and C. Franco-Paredes, The expanding spectrum of modes of transmission of Zika virus: A global concern, Ann. Clin. Microbiol. Antimicrob. , 15 (2016), 13, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4776405/.Google Scholar

[21]

R. Ross, The Prevention of Malaria, Churchill, London, 1911.Google Scholar

[22]

Q. Zhang, K. Sun, M. Chinazzi, A. Pastore-Piontti, N. E. Dean, D. P. Rojas, S. Merler, D. Mistry, P. Poletti, L. Rossi, M. Bray, M. E. Halloran, I. M. Longini and A. Vespignani, Spread of Zika virus in the Americas, Proc. Natl. Acad. Sci. (2017). doi: 10.1073/pnas.1620161114. Google Scholar

show all references

References:
[1]

C. L. Althaus and N. Low, How Relevant Is Sexual Transmission of Zika Virus?, PLOS Med., 13 (2016), 1-3. doi: 10.1371/journal.pmed.1002157. Google Scholar

[2]

M.-C. BoilyR. F. BaggaleyL. WangB. MasseR. G. WhiteR. J. Hayes and M. Alary, Heterosexual risk of HIV-1 infection per sexual act: Systematic review and meta-analysis of observational studies, Lancet Infect. Dis., 9 (2009), 118-129. doi: 10.1016/S1473-3099(09)70021-0. Google Scholar

[3]

R. B. BrooksM. P. CarlosR. A. MyersM. Grace WhiteT. Bobo-LenociD. AplanD. Blythe and K. A. Feldman, Likely Sexual Transmission of Zika Virus from a Man with No Symptoms of Infection --Maryland, 2016, MMWR Morb Mortal Wkly Rep, 65 (2016), 915-916. doi: 10.15585/mmwr.mm6534e2. Google Scholar

[4]

CDC, Hepatitis A Questions and Answers for the Public, URL https://www.cdc.gov/hepatitis/hav/afaq.htm.Google Scholar

[5]

F. C. CoelhoB. DurovniV. SaraceniC. LemosC. T. CodecoS. CamargoL. M. de CarvalhoL. BastosD. ArduiniD. A. M. Villela and M. Armstrong, 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., 51 (2016), 128-132. doi: 10.1016/j.ijid.2016.08.023. Google Scholar

[6]

W. D. DavidsonS SlavinskiK. Komoto and J. Rakeman, Suspected Female-to-Male Sexual Transmission of Zika Virus-New York City, 2016, MMWR Morb Mortal Wkly Rep, 65 (2016), 716-717. Google Scholar

[7]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, New York: John Wiley and Sons, 2000. Google Scholar

[8]

M. R. DuffyT.-H. ChenW. T. HancockA. M. PowersJ. L. KoolR. S. LanciottiM. PretrickM. MarfelS. HolzbauerC. DubrayL. GuillaumotA. GriggsM. BelA. J. LambertJ. LavenO. KosoyA. PanellaB. J. BiggerstaffM. Fischer and E. B. Hayes, Zika virus outbreak on Yap Island, Federated States of Micronesia., N. Engl. J. Med., 360 (2009), 2536-2543. Google Scholar

[9]

N. M. Ferguson, Z. M. Cucunubá, I. Dorigatti, G. L. Nedjati-Gilani, C. A. Donnelly, M.- G. Basáñez, P. Nouvellet and J. Lessler, Countering the Zika epidemic in Latin America, Science (80-. )., 353 (2016), 353-354, URL http://www.sciencemag.org/cgi/content/full/science.aag0219/DC1.Google Scholar

[10]

K. Fonseca, B. Meatherall, D. Zarra, M. Drebot, J. MacDonald, K. Pabbaraju, S. Wong, P.Webster, R. Lindsay and R. Tellier, First case of zika virus infection in a returning canadian traveler, Am. J. Trop. Med. Hyg., 91 (2014), 1035-1038, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4228871/.Google Scholar

[11] P. J. Fos, Epidemiology Foundations: The Science of Public Health, Jossey-Bass, San Francisco, 2011. Google Scholar
[12]

E. B. Hayes, Zika Virus Outside Africa, Emerg. Infect. Dis., 15 (2009), 1347-1350, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2819875/http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2819875/.Google Scholar

[13]

W. O. Kermack and A. G. McKendrick, A contributions to the mathematical theory of epidemics, Proc. R. Soc. London A, 115 (1927), 700-721. Google Scholar

[14]

A. J. KucharskiS. FunkR. M. EggoH.-P. MalletW. J. Edmunds and E. J. Nilles, Transmission dynamics of zika virus in island populations: A modelling analysis of the 2013-14 french polynesia outbreak, PLoS Negl. Trop. Dis, 10 (2016), e0004726. doi: 10.1371/journal.pntd.0004726. Google Scholar

[15]

W. D. Z. Lopes, J. D. Rodriguez, F. A. Souza, T. R. dos Santos, R. S. dos Santos, W. M. Rosanese, W. R. Z. Lopes, C. A. Sakamoto and A. J. da Costa, Sexual transmission of Toxoplasma gondii in sheep, Vet. Parasitol., 195 (2013), 47-56, URL http://linkinghub.elsevier.com/retrieve/pii/S0304401713000083.Google Scholar

[16]

G. MacDonald, The analysis of equilibrium in malaria, Trop Dis Bull, 49 (1952), 813-829. Google Scholar

[17]

J. M. MansuyC. PasquierM. DaudinS. Chapuy-RegaudN. MoinardC. ChevreauJ. IzopetC. Mengelle and L. Bujan, Zika virus in semen of a patient returning from a non-epidemic area, Lancet Infect. Dis., 16 (2016), 894-895. doi: 10.1016/S1473-3099(16)30153-0. Google Scholar

[18]

J. C. Miller, Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes, Infect. Dis. Model., 2 (2017), 35-55, URL http://linkinghub.elsevier.com/retrieve/pii/S2468042716300203.Google Scholar

[19]

R. Pellissier and A. Rousselot, Enquete serologique sur l'incidence des virus neurol ropes chez quelques singes de l'Afrique Equatoriale Francaise (French) [A Serological Investigation of the Incidence of Neurotropic Viruses in Certain Monkeys of French Equatorial Africa], Bull. Société Pathol. Exot., 47 (1954), 228-231. Google Scholar

[20]

A. J. Rodriguez-Morales, A. C. Bandeira and C. Franco-Paredes, The expanding spectrum of modes of transmission of Zika virus: A global concern, Ann. Clin. Microbiol. Antimicrob. , 15 (2016), 13, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4776405/.Google Scholar

[21]

R. Ross, The Prevention of Malaria, Churchill, London, 1911.Google Scholar

[22]

Q. Zhang, K. Sun, M. Chinazzi, A. Pastore-Piontti, N. E. Dean, D. P. Rojas, S. Merler, D. Mistry, P. Poletti, L. Rossi, M. Bray, M. E. Halloran, I. M. Longini and A. Vespignani, Spread of Zika virus in the Americas, Proc. Natl. Acad. Sci. (2017). doi: 10.1073/pnas.1620161114. Google Scholar

Figure 1.  attack ratios $z_1, z_2$ obtained for model (1) with $\mathcal{R}_0 = 1.25$ (left panel) and $1.5$ (right panel). The $x$-axis displays (in logarithmic scale) the quantity $\frac{\beta_f \gamma_f}{ \beta_m \gamma_m}$ (which we called relative susceptibility of female to male, even though the recovery rates are also included), and the $y$-axis displays the corresponding values $\bar z=(\bar z_1, \bar z_2)$ solving $H(z) = 0$, as well as their average $(\bar z_1+\bar z_2)/2$
Figure 2.  Ratio between the sex-specific final attack ratios, $\bar z_f/\bar z_m$, as a function of relative susceptibility $\frac{\beta_f }{ \beta_m }$ for different values of $\mathcal{R}_0$. The black curves are obtained using the model (1) including heterosexual transmission only and $\mathcal{R}_0$ is given by (6); the red curves using the model (7) that includes both types of transmission, where $\mathcal{R}_0$ is given by (19) and $\rho$ defined in (21) equal to 50 %. Here, for the sake of simplicity, we have set $\gamma_m = \gamma_f$
Figure 3.  Contour plot of ratios of sex-specific final attack ratios, $\bar z_f/\bar z_m$, as a function of relative susceptibility $\frac{\beta_f }{ \beta_m }$ and $\rho$ defined in (21). Here $\mathcal{R}_0^{sn} = 1.5$
Figure 4.  Ratio of sex-specific final attack ratios, $\bar z_f/\bar z_m$, as a function of average attack ratio $(\bar z_f + \bar z_m)/2$ for models (7) (lines) and (22) for different values of $q$ (see legend). Here $\rho = 1/2$ and parameters are varied to keep $q$ and $\rho$ at these values
Figure 5.  One simulation of model (22). Long-dashed line represents infected females, dotted line infected males; solid line is ratio $I_f(t)/I_m(t)$. Parameter values are $\beta_f =0.442$, $\beta_m = 0.0442$, $\beta_V = 0.05$, $\beta_H = 0.035$, $\gamma = 1/6$, $\mu_V = 1/5$, $N =1 \times 10^4$, $V=5.35 \times 10^5$, so that $\mathcal{R}_0 = 1.8$ using (24), while $\mathcal{R}_0^s = \mathcal{R}_0^v = 0.7$, and final attack ratios are $z_m=0.76$, $z_f = 0.96$
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