# Advanced Hepatitis C Virus Replication PDE Models within a Realistic Intracellular Geometric Environment

^{1}

Department of Mathematics, Chair of Applied Mathematics 1, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany

^{2}

Medivir AB, Department of Biology, 141 22 Huddinge, Sweden

^{3}

Dipartimento di Scienze Matematiche (DISMA) “G.L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino (TO), Italy

^{4}

Department of Medicine, Institute for Biostatistics and Mathematic Modeling, Goethe Universität Frankfurt, Theodor-Stern-Kai 7, 60590 Frankfurt am Main, Germany

^{5}

Goethe Center for Scientific Computing (G-CSC), Goethe Universität Frankfurt, Kettenhofweg 139, 60325 Frankfurt am Main, Germany

^{6}

Applied Mathematics and Computational Science, King Abdullah University of Science and Technology (KAUST), 23955-6900 Thuwal, Saudi Arabia

^{*}

Author to whom correspondence should be addressed.

^{†}

Affiliation 5 was affiliation of M.M.K until March 2018. The major part of this study was performed at the G-CSC.

^{‡}

These authors contributed equally to this work.

*Int. J. Environ. Res. Public Health*

**2019**,

*16*(3), 513; https://0-doi-org.brum.beds.ac.uk/10.3390/ijerph16030513

Received: 29 October 2018 / Revised: 8 January 2019 / Accepted: 16 January 2019 / Published: 12 February 2019

(This article belongs to the Special Issue Spatio-Temporal Analysis of Infectious Diseases)

The hepatitis C virus (HCV) RNA replication cycle is a dynamic intracellular process occurring in three-dimensional space (3D), which is difficult both to capture experimentally and to visualize conceptually. HCV-generated replication factories are housed within virus-induced intracellular structures termed membranous webs (MW), which are derived from the Endoplasmatic Reticulum (ER). Recently, we published 3D spatiotemporal resolved diffusion–reaction models of the HCV RNA replication cycle by means of surface partial differential equation (sPDE) descriptions. We distinguished between the basic components of the HCV RNA replication cycle, namely HCV RNA, non-structural viral proteins (NSPs), and a host factor. In particular, we evaluated the sPDE models upon realistic reconstructed intracellular compartments (ER/MW). In this paper, we propose a significant extension of the model based upon two additional parameters: different aggregate states of HCV RNA and NSPs, and population dynamics inspired diffusion and reaction coefficients instead of multilinear ones. The combination of both aspects enables realistic modeling of viral replication at all scales. Specifically, we describe a replication complex state consisting of HCV RNA together with a defined amount of NSPs. As a result of the combination of spatial resolution and different aggregate states, the new model mimics a cis requirement for HCV RNA replication. We used heuristic parameters for our simulations, which were run only on a subsection of the ER. Nevertheless, this was sufficient to allow the fitting of core aspects of virus reproduction, at least qualitatively. Our findings should help stimulate new model approaches and experimental directions for virology.

*Keywords:*computational virology; hepatitis C virus (HCV); viral dynamics; within-host viral modeling; mathematical models of viral RNA cycle; population dynamics; 3D spatiotemporal resolved mathematical models; realistic geometries; (surface) partial differential equations; Finite Volumes; massively parallel multigrid solvers