A sexually transmitted infection model with long-term partnerships in homogeneous and heterogenous populations

Abstract

Population models for sexually transmitted infections frequently use a transmission model that assumes an inherent partnership length of zero. However, in a population with long-term partnerships, the infection status of the partners, the length of the partnership, and the exclusivity of the partnership significantly affect the rate of infection. We develop an autonomous population model that can account for the possibilities of an infection from either a casual sexual partner or a longtime partner who was either infected at the start of the partnership or was newly infected. The impact of the long-term partnerships on the rate of infection is captured by calculating the expected values of the rate of infection from these extended contacts. We present a new method to evaluate partner acquisition rates for casual or long-term partnerships which produces in a more realistic number of lifetime sexual partners. Results include a SI model with different infectiousness levels for the transmission of HIV and HSV-2 with acute and chronic/latent infection stages for homogeneous (MSM) and heterogeneous (WSM-MSW) groups. The accompanying reproduction number and sensitivity studies highlight the impact of both casual and long-term partnerships on infection spread. We construct an autonomous set of equations that handle issues usually ignored by autonomous equations and handled only through simulations or in a non-autonomous form. The autonomous formulation of the model allows for simple numerical computations while incorporating a combination of random instantaneous contacts between individuals and prolonged contacts between specific individuals.

Fig. 1

$S{I}_1{I}_2$ model, where S is the susceptible population, $I_1$ is the acutely infected population, $I_2$ is the chronic or latent population.

Fig. 2

Heterogeneous $S{I}_1{I}_2$ model with populations divided into two categories, W and M.

Fig. 3

Test of the effect of the including long-term partnership information into the model on the reproduction numbers. All parameters are evaluated at the baseline HIV and HSV-2 values shown in Table 1, Table 2. Figure (a) represents MSM data and Figure (b) WSM-MSW data.

Fig. 4

Test of the effect of the long-term partnership duration and the rate of acquiring long-term and casual partners on the reproduction numbers evaluated at baseline HIV parameter values shown in Table 1, Table 2 with the exception of τ and ξ. Both figures use $\xi =10\text{\%}$. The p and z values were chosen to keep the average number of partners over one year to be $4/3$. The vertical scale is different for (a) and (b) to best illustrate the effect of different p, z, and τ values for the different population groups. Figure (a) represents MSM results and (b) WSM-MSW results.

Fig. 5

Reproduction number for WSM-MSW where the total number of sex acts per year is held constant.

Fig. 6

Test of the effect of the long-term partnership duration and the rate of acquiring long-term and casual partners on the reproduction numbers evaluated at baseline HSV-2 parameter values shown in Table 1, Table 2 with the exception of τ and ξ. All figures use $\xi =10\text{\%}$. The $p=p_M=p_W$ and $z=z_M=z_W$ values were chosen to keep the average number of partners over one year to be $4/3$. The vertical scale is different for (a) and (b) to best illustrate the effect of different p, z, and τ values for the different population groups. Figure (a) represents MSM results and (b) WSM-MSW results.