Effects of contact structure on the transient evolution of HIV virulence

Abstract

Early in an epidemic, high densities of susceptible hosts select for relatively high parasite virulence; later in the epidemic, lower susceptible densities select for lower virulence. Thus over the course of a typical epidemic the average virulence of parasite strains increases initially, peaks partway through the epidemic, then declines again. However, precise quantitative outcomes, such as the peak virulence reached and its timing, may depend sensitively on epidemiological details. Fraser et al. proposed a model for the eco-evolutionary dynamics of HIV that incorporates the tradeoffs between transmission and virulence (mediated by set-point viral load, SPVL) and their heritability between hosts. Their model used implicit equations to capture the effects of partnership dynamics that are at the core of epidemics of sexually transmitted diseases. Our models combine HIV virulence tradeoffs with a range of contact models, explicitly modeling partnership formation and dissolution and allowing for individuals to transmit disease outside of partnerships. We assess summary statistics such as the peak virulence (corresponding to the maximum value of population mean log10 SPVL achieved throughout the epidemic) across models for a range of parameters applicable to the HIV epidemic in sub-Saharan Africa. Although virulence trajectories are broadly similar across models, the timing and magnitude of the virulence peak vary considerably. Previously developed implicit models predicted lower virulence and slower progression at the peak (a maximum of 3.5 log10 SPVL) compared both to more realistic models and to simple random-mixing models with no partnership structure at all (both with a maximum of ≈ 4.7 log10 SPVL). In this range of models, the simplest random-mixing structure best approximates the most realistic model; this surprising outcome occurs because the dominance of extra-pair contact in the realistic model swamps the effects of partnership structure.

Fig 1. Baseline dynamics.

Time series of mean population log₁₀ SPVL using baseline parameters. (a) Contrast between the three-stage Shirreff model and the single-stage model calibrated to varying initial exponential growth rates, r. (b) Effects of varying initial infectious density I(0). (c) Effects of varying initial mean virulence α(0). The middle r = 0.042 (orange, dotted) curve in panel (a), calibrated to match the epidemic dynamics of Shirreff et al.’s three-stage model [9], shows that our simplified single-stage HIV model can produce similar SPVL trajectories to the original three-stage model; the other r values are chosen to show the effects of doubling or halving the initial epidemic growth rate. The other panels (b,c) show that eco-evolutionary dynamics are qualitatively similar across a range of initial conditions.

Fig 2. Model contact structures.

Schematic representations of models with explicit contact structure. Top row (a,b) shows models with instantaneous partnership formation; bottom row (c,d) shows those where individuals remain single for some period of time between partnerships. Left column (a,c) shows models with within-partnership transmission only; in these models transmission can only occur within serodiscordant partnerships, and serodiscordant partnerships can only be formed by dissolution and reformation of partnerships. Right column (b,d) shows models that allow extra-pair and uncoupled transmission. Solid arrows represent infection transitions; dotted arrows represent partnership dissolution and formation; dashed arrows show influences on infection rate, including both within-pair transmission (blue) and extra-pair/uncoupled transmission (red). In addition to these models, we also use random-mixing and implicit models with only S and I compartments and heterogeneous models where each compartment is subdivided by sexual activity rate (of both partners, in the case of partnership compartments).

Fig 3. Envelopes of virulence trajectories (population mean log₁₀ SPVL) under all models.

Panels are arranged in order of complexity of contact structure, from “random” (least complex) to “hetero” (most complex). All models were run until t = 4000 years using parameters from 1000 Latin hypercube samples to illustrate the range of possible dynamics. Envelopes contain the middle 95% of trajectories (i.e. we select all points between the 0.025 and 0.975 quantiles for each model at each year), while center lines show mean trajectories. Truncated series (up to year 800) are shown here.

Fig 4. Univariate distributions of summary statistics.

Models arranged in order of complexity, as in Fig 3. For every simulation, three summary statistics are shown: peak time, maximum mean log₁₀ SPVL, and equilibrium mean log₁₀ SPVL. Peak time is the time at which the SPVL reaches its maximum for each simulation run (maximum mean log₁₀ SPVL). Equilibrium mean log₁₀ SPVL is taken by taking the mean log₁₀ SPVL value at t = 4000. Because the distribution of equilibrium mean log₁₀ SPVL (lower left panel) for the random-mixing model is very narrow, it has been replaced by a point in order to preserve the vertical axis scaling.

Fig 5. Pairs plot: bivariate relationships among summary statistics for each model structure.

The three summary statistics for each simulation are plotted against each other in order to visualize the relationship among the summary statistics and to help compare models. Surprisingly, the implicit model, an approximation for instantaneous partnership formation model (instswitch), shows an almost identical trend with a model that has pair formation dynamics (pairform). The dashed line in the equilibrium vs. peak virulence plot (lower left) shows the 1:1 line, where equilibrium and peak virulence are equal. To avoid too much overplotting, only 10% of the parameter sets (randomly sampled) are shown here.

Fig 6. Sensitivity plot.

Distribution (points) and trend (smooth line) of summary statistics (y-axis, rows) as a function of parameter values (x-axis, columns), some of which have been scaled to calibrate the initial epidemic growth rate. Changing parameters may have differing effects for models with different contact structures: for example, partnership dissolution rate (c, third column) increases peak time and virulence in models without explicit pair formation but decreases them in more realistic models.