Modeling Viral Spread

Abstract

The way in which a viral infection spreads within a host is a complex process that is not well understood. Different viruses, such as human immunodeficiency virus type 1 and hepatitis C virus, have evolved different strategies, including direct cell-to-cell transmission and cell-free transmission, to spread within a host. To what extent these two modes of transmission are exploited in vivo is still unknown. Mathematical modeling has been an essential tool to get a better systematic and quantitative understanding of viral processes that are difficult to discern through strictly experimental approaches. In this review, we discuss recent attempts that combine experimental data and mathematical modeling in order to determine and quantify viral transmission modes. We also discuss the current challenges for a systems-level understanding of viral spread, and we highlight the promises and challenges that novel experimental techniques and data will bring to the field.

Keywords: HCV; HIV; agent-based models; cell-free virus infection; cell-to-cell infection; mathematical modeling.

Figure 1

(a) Sketch of the mathematical model given by Equation 1, describing viral spread by cell-to-cell (red ) and cell-free (black) transmission. (b) Dynamics of uninfected (solid line) and infected (dashed line) cells (upper panel ), and corresponding dynamics of cell-free virus (lower panel ), for realizations of the model allowing for either cell-to-cell and cell-free transmission (black) or only cell-free transmission (red ). Parameters used in the model were λ = 100 day⁻¹, d = 0.1 day⁻¹, ρ = 1.5 × 10³ day⁻¹, c = 10 day⁻¹, δ = 0.5 day⁻¹, β = 10⁻⁵ day⁻¹ per virion, and ω = 10⁻³ day⁻¹ per infected cell.

Figure 2

Agent-based model of hepatitis C virus cell-to-cell spread. (a) Cells are modeled as hexagonal structures within a regular grid and can change their status, x_{i ,j}(t), with time. Uninfected cells (white hexagons) get infected ( green hexagons) by cell-free virus ( purple stars) with probability p_cf or by cell-to-cell transmission from direct neighbors that are infectious (red hexagons) with probability p_cc. Intracellular viral replication and viral export are explicitly modeled, distinguishing between positive-strand vRNA (H) and replication complexes (R). Positive-strand vRNA is responsible for cell-to-cell transmission and, when exported in viral particles, for cell-free virus transmission. Cell-free virus can diffuse throughout the grid. Infection is initiated by introducing a limited number of infected cells onto the grid. (b) Output from two simulations assuming either cell-free and cell-to-cell transmission (left) or only cell-to-cell transmission (right). Simulations comprise ~10,000 cells in total, and snapshots of parts of the simulated cell culture are shown when a total of ~300 cells were infected. The black hexagons indicate the initially infected cell that founded the corresponding large infected cell focus.

Figure 3

Current challenges for the analysis of viral spread. (a) The number of viral particles transmitted per contact, m, and how the magnitude of m affects viral dynamics and evolution are not known. (b) The kinetics of contact formation and duration of contact needed to successfully infect other cells are also not known in the case of motile cells, such as CD4⁺ T cells. (c) The dynamics and protective capacity of local innate immune responses triggered by infection, such as type I interferon (IFN) responses, still need to be characterized for most viral infections. How quickly agents such as IFN released by infected and nearby cells (e.g., plasmacytoid dendritic cells or target cells in which viral products have been sensed but that have not yet become productively infected) render neighboring cells protected against infection is still an open question. Quantification of these processes is needed in order to advance the analysis of viral spread. Red cells are infectious, orange cells are becoming infected, blue cells are protected, and gray cells are uninfected. Blue arrows indicate release of IFN, and black arrows with red crosses indicate the spread of infection being blocked due to the protected state of blue cells.