Minimal within-host dengue models highlight the specific roles of the immune response in primary and secondary dengue infections

Abstract

In recent years, the within-host viral dynamics of dengue infections have been increasingly characterized, and the relationship between aspects of these dynamics and the manifestation of severe disease has been increasingly probed. Despite this progress, there are few mathematical models of within-host dengue dynamics, and the ones that exist focus primarily on the general role of immune cells in the clearance of infected cells, while neglecting other components of the immune response in limiting viraemia. Here, by considering a suite of mathematical within-host dengue models of increasing complexity, we aim to isolate the critical components of the innate and the adaptive immune response that suffice in the reproduction of several well-characterized features of primary and secondary dengue infections. By building up from a simple target cell limited model, we show that only the innate immune response is needed to recover the characteristic features of a primary symptomatic dengue infection, while a higher rate of viral infectivity (indicative of antibody-dependent enhancement) and infected cell clearance by T cells are further needed to recover the characteristic features of a secondary dengue infection. We show that these minimal models can reproduce the increased risk of disease associated with secondary heterologous infections that arises as a result of a cytokine storm, and, further, that they are consistent with virological indicators that predict the onset of severe disease, such as the magnitude of peak viraemia, time to peak viral load, and viral clearance rate. Finally, we show that the effectiveness of these virological indicators to predict the onset of severe disease depends on the contribution of T cells in fuelling the cytokine storm.

Keywords: cytokine storm; dengue; disease severity; immune response; mathematical model; viral dynamics.

Figure 1.

Primary infection model analysis. Rows show distinct within-host models examined; columns show the three characteristic features that the models seek to reproduce. The models are: (a–c) target cell limited model (equations (2.2)); (d–f) innate immune response model with fast NK-cell dynamics (equations (2.4)) and (g–i) innate immune response model with explicit NK-cell dynamics (equations (2.5)). The characteristic features are: (a,d,g) the final fraction of uninfected cells; (b,e,h) time to peak viraemia (in days) and (c,f,i) daily maximum viral clearance rate (in log₁₀ copies ml⁻¹ d⁻¹). Each subplot shows the values of one characteristic feature as a function of two varied model parameters. Values in yellow meet the desired feature values of a primary dengue infection. Arrows point to the parameter values for which two or three characteristic features are simultaneously met. In subplots (a–c), I(0) = 1.4 × 10⁻⁴⁷ cells ml⁻¹. In subplots (d–i), I(0) = 8.62 × 10⁻¹⁸ cells ml⁻¹. The target cell limited model meets all characteristic dengue features when βρ = 7.9 × 10⁻⁵ (cell ml⁻¹)⁻¹ d⁻¹ and δ = 21.7 per day, although this high δ value is biologically implausible.

Figure 2.

Within-host dynamics of primary and secondary dengue infections. Primary infection dynamics are shown in black solid lines; secondary infection dynamics are shown in grey dashed lines in subplots (a–e). (a) The fraction of uninfected cells (S(t)/S(0)) over the course of an infection, in days. (b) Infected cell dynamics (I(t); in cells ml⁻¹). (c) Viral load dynamics (V(t); in copies ml⁻¹). (d) NK-cell dynamics (cells ml⁻¹). (e) T-cell dynamics (cells ml⁻¹), secondary infection only. (f) The dynamics of endothelial activators (pg ml⁻¹). For endothelial activator dynamics in dotted grey, it is assumed that endothelial activators are secreted by both infected cells and T cells (α = 2 × 10⁻⁵). For endothelial activator dynamics in dashed grey, it is assumed endothelial activators are only secreted by infected cells (α = 0).

Figure 3.

Secondary infection model analysis. The effect of changes in parameters q_T and δ_T on features of a secondary dengue infection using model (2.6): (a) peak viraemia value (log₁₀ copies ml⁻¹); (b) time to peak viraemia (days); and (c) daily maximum viral clearance rate (log₁₀ copies ml⁻¹ d⁻¹). Values in yellow meet the desired feature values of a secondary dengue infection. Arrows point to the parameter values for which all characteristic features are met.

Figure 4.

Simulations of the minimal models, parametrized according to table 1, alongside viral load data from DENV-1-infected adult patients [25]. (a) Primary infection data and model dynamics. Viral load data from primary infection DF patients (n = 15) are shown in green; viral load data from primary infection DHF patients (n = 3) are shown in grey. (b) Secondary infection data and model dynamics. Viral load data from secondary infection DF patients (n = 91) are shown in green; viral load data from secondary infection DHF patients (n = 32) are shown in grey. Model simulations recover generally shorter times to peak viraemia [9,56], higher viral clearance rates, [9,11] and higher peak viraemia levels [9,29] observed in symptomatic secondary infections relative to symptomatic primary infections.

Figure 5.

The relationship between virological indicators and the risk of developing severe disease. S(0), β, ρ, κ, q_N(q/d) and d_N are varied for primary and secondary infections. T(0), δ_T, q_T and d_T are additionally varied for a secondary infection. Primary infection simulations are shown with pink dots. Secondary infection simulations are shown with dark blue pluses (α = 0) and light blue diamonds (α = 2 × 10⁻⁵). (a) A scatterplot of peak viraemia (log₁₀ copies ml⁻¹) and the maximum value of endothelial activators (pg ml⁻¹) for each LHS simulation. (b) A scatterplot of time to viral peak (days) and the maximum value of endothelial activators (pg ml⁻¹) for each LHS simulation. (c) A scatterplot of daily viral clearance rate (log₁₀ copies ml⁻¹ d⁻¹) and the maximum value of endothelial activators (pg ml⁻¹) for each LHS simulation.