A hybrid PDE–ABM model for viral dynamics with application to SARS-CoV-2 and influenza

Abstract

We propose a hybrid partial differential equation–agent-based (PDE–ABM) model to describe the spatio-temporal viral dynamics in a cell population. The virus concentration is considered as a continuous variable and virus movement is modelled by diffusion, while changes in the states of cells (i.e. healthy, infected, dead) are represented by a stochastic ABM. The two subsystems are intertwined: the probability of an agent getting infected in the ABM depends on the local viral concentration, and the source term of viral production in the PDE is determined by the cells that are infected. We develop a computational tool that allows us to study the hybrid system and the generated spatial patterns in detail. We systematically compare the outputs with a classical ODE system of viral dynamics, and find that the ODE model is a good approximation only if the diffusion coefficient is large. We demonstrate that the model is able to predict SARS-CoV-2 infection dynamics, and replicate the output of in vitro experiments. Applying the model to influenza as well, we can gain insight into why the outcomes of these two infections are different.

In Silico Evaluation of Paxlovid’s Pharmacometrics for SARS-CoV-2: A Multiscale Approach

“…One of the most important modeling decisions in [ 6 ] was defining epithelial cells as discrete agents in an agent-based model (ABM). The diameter of epithelial cells is relatively significant [ 11 , 12 ] and consequently, in terms of mathematical conceptualization, it is natural to approach cells as separate entities and follow their respective states on an individual level.…”

“…The discrete state space of target cells is defined precisely as in [ 6 ]. For completeness, we recall some of the fundamental technical details; namely, we construct a two-dimensional ABM state space by introducing a lattice of agents representing epithelial cells ( ).…”

“…Concerning state dynamics, the transition rules are set to mimic the biological phenomenon in question, the complete list is as follows: All living uninfected cells are susceptible target cells to virus infection; Since the time frame of infection is relatively short, cell birth and cell division are ignored; Infection is not reversible: an infected cell can not become a healthily functioning uninfected cell again; Viral infection itself is the only reason for cell death, i.e., death related to any other natural cause is not accounted for (considering the 17-month half-life of lung epithelial cells [ 13 ], natural apoptosis may be ignored over the course of a 5-day Paxlovid treatment); The uninfected → infected state change: a target cell may become infected depending on the local virus concentration at the given cell. Infection itself is randomized and it occurs with a probability of (for more details, see [ 6 ]); The infected → dead state change: Analogously to infection, death is governed by a stochastic model with probability . …”

“…More formally, building on our previous work [ 6 ], we define a hybrid mathematical model by merging (i) a partial differential equation representing local virus concentration, (ii) an agent-based model describing target cells in the lung and their three possible states (uninfected, infected, and dead), and (iii) a partial differential equation representing nirmatrelvir concentration. Naturally, the respective parts are closely and meaningfully intertwined: each considerable interaction and feedback process connecting these separate biological participants is given formal definition and appears in the implementation.…”

“…Naturally, the respective parts are closely and meaningfully intertwined: each considerable interaction and feedback process connecting these separate biological participants is given formal definition and appears in the implementation. A detailed motivation, definition and construction of this model type and its various advantages are discussed in [ 6 ]—here, we limit ourselves to highlighting the role of including crucial spatial mechanisms: unlike other classical models such as the ODE (ordinary differential equations) approach, the present system is defined in both space and time, and consequently, it is able to capture highly significant—and in nature inherently spatial—physical phenomena such as virus diffusion.…”