# American Institute of Mathematical Sciences

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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
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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 $(\bar z_1+\bar z_2)/2$
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 $\gamma_m = \gamma_f$
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 $\mathcal{R}_0^{sn} = 1.5$
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
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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