Ebola virus infection modeling and identifiability problems

Abstract

The recent outbreaks of Ebola virus (EBOV) infections have underlined the impact of the virus as a major threat for human health. Due to the high biosafety classification of EBOV (level 4), basic research is very limited. Therefore, the development of new avenues of thinking to advance quantitative comprehension of the virus and its interaction with the host cells is urgently needed to tackle this lethal disease. Mathematical modeling of the EBOV dynamics can be instrumental to interpret Ebola infection kinetics on quantitative grounds. To the best of our knowledge, a mathematical modeling approach to unravel the interaction between EBOV and the host cells is still missing. In this paper, a mathematical model based on differential equations is used to represent the basic interactions between EBOV and wild-type Vero cells in vitro. Parameter sets that represent infectivity of pathogens are estimated for EBOV infection and compared with influenza virus infection kinetics. The average infecting time of wild-type Vero cells by EBOV is slower than in influenza infection. Simulation results suggest that the slow infecting time of EBOV could be compensated by its efficient replication. This study reveals several identifiability problems and what kind of experiments are necessary to advance the quantification of EBOV infection. A first mathematical approach of EBOV dynamics and the estimation of standard parameters in viral infections kinetics is the key contribution of this work, paving the way for future modeling works on EBOV infection.

Keywords: EBOV; Ebola; identifiability; kinetics; mathematical modeling; viral dynamics.

Figure 1

Ebola virus molecular structure. The Ebola genome is composed of 3 leader, nucleoprotein (NP), virion protein 35 (VP35), VP40, glycoprotein (GP), VP30, VP24, polymerase (L) protein and 5 trailer (adapted from SIB SWISS Institute of Bioinformatics, 2014).

Figure 2

Schematic representation of the model for EBOV infection. Target cells (U) are replenished with rate λ and die with rate ρ. Virus (V) infects target cells (U) with rate β. Infected cells are cleared with rate δ. Once cells are productively infected (I), they release virus at rate p and virus particles are cleared with rate c.

Figure 3

Data preparation. Fitted statistical model for the wild-type Vero cells infected with EBOV at a low multiplicity of infection (MOI) (Halfmann et al., 2008)

Figure 4

Parameter Identifiability. RMS profile of model parameters. Each parameter is varied in a wide range around the optimized value. Subsequently, the DE algorithm is used to refit the remaining parameters to the data set of Halfmann et al. (2008). The vertical dashed lines indicate the value obtained from the optimization for all four parameters collectively.

Figure 5

Weighted bootstrap results. Top row: Distributions from 1000 sample estimates are presented for the three parameters: β, p and c. Bottom row: Scatter plot between bootstrap parameters. The parameter ρ is fixed during the bootstrapping at 0.001 (Moehler et al., 2005). Numerical values for the model Equations (1–3) are presented in the Table 1.

Figure 6

Sensitivity of parameters. (A–E) Plotting of viral titer variation vs. time. The dashed line is the viral kinetics obtain from nominal parameter values. Three color shades in each figure represent the viral load variation range when varying the corresponding parameter by a percentage denoted in the legend. (F) Parameters sensitivity function over time, the values in y-axis are calculated using Equation (5).

Figure 7

Model fitting for EBOV kinetics. Viral titer data with low MOI from Halfmann et al. (2008) and simulations from the best fit shown in Table 1 are in panel (A) for the host cells and (B) for the viral titer.

Figure 8

Transmission measures. Bootstrap estimate of (A) reproductive number and (B) infecting time in hours. Numerical values can be found in Table 1.

Figure 9

Cross-validation. Test of estimated parameters on an independent set of data. The viral replication kinetics in wild-type Vero cells infected with EBOV at a high multiplicities of infection (MOI) in Halfmann et al. (2008) are modeled starting from a higher initial viral load of V₀ = 460 ffu/ml. The (Mean) indicates the predicted kinetics using parameters obtained from bootstrap while (Best) refers to the predicted kinetics using the parameters resulting from the optimization.