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.
Highlights
Mathematical models have been powerful tools in tackling the challenges posed by the appearance of the COVID-19 disease caused by the novel SARS-CoV-2 coronavirus
The first model we considered was a hybrid PDE–agent-based model (ABM) system, which is essentially a result of merging a discrete state space representing epithelial cells with a continuous reaction–diffusion equation grasping virus concentration
As for theoretical completeness, we provide a rigorous analysis of both models in the appendices, including a wellposedness result related to the hybrid model and the study of the ODE model’s temporal dynamics
Summary
Mathematical models have been powerful tools in tackling the challenges posed by the appearance of the COVID-19 disease caused by the novel SARS-CoV-2 coronavirus. In the ongoing and virtually unprecedented pandemic, these mathematical models are invaluable as they are able to provide insights or predictions based on mathematical analysis and computer simulations. This means that their results are obtained at an ideally low cost even in complex situations where real-life and real-time experiments to obtain these same results would be royalsocietypublishing.org/journal/rsos R. There is a large variety of both large- and small-scale 2 mathematical models related to COVID-19. We use a hybrid mathematical approach for our study, focusing on SARSCoV-2 and influenza infections
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