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Zitzmann C, Dächert C, Schmid B, van der Schaar H, van Hemert M, Perelson AS, van Kuppeveld FJ, Bartenschlager R, Binder M, Kaderali L. Mathematical modeling of plus-strand RNA virus replication to identify broad-spectrum antiviral treatment strategies. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2022:2022.07.25.501353. [PMID: 35923314 PMCID: PMC9347285 DOI: 10.1101/2022.07.25.501353] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
Abstract
Plus-strand RNA viruses are the largest group of viruses. Many are human pathogens that inflict a socio-economic burden. Interestingly, plus-strand RNA viruses share remarkable similarities in their replication. A hallmark of plus-strand RNA viruses is the remodeling of intracellular membranes to establish replication organelles (so-called "replication factories"), which provide a protected environment for the replicase complex, consisting of the viral genome and proteins necessary for viral RNA synthesis. In the current study, we investigate pan-viral similarities and virus-specific differences in the life cycle of this highly relevant group of viruses. We first measured the kinetics of viral RNA, viral protein, and infectious virus particle production of hepatitis C virus (HCV), dengue virus (DENV), and coxsackievirus B3 (CVB3) in the immuno-compromised Huh7 cell line and thus without perturbations by an intrinsic immune response. Based on these measurements, we developed a detailed mathematical model of the replication of HCV, DENV, and CVB3 and show that only small virus-specific changes in the model were necessary to describe the in vitro dynamics of the different viruses. Our model correctly predicted virus-specific mechanisms such as host cell translation shut off and different kinetics of replication organelles. Further, our model suggests that the ability to suppress or shut down host cell mRNA translation may be a key factor for in vitro replication efficiency which may determine acute self-limited or chronic infection. We further analyzed potential broad-spectrum antiviral treatment options in silico and found that targeting viral RNA translation, especially polyprotein cleavage, and viral RNA synthesis may be the most promising drug targets for all plus-strand RNA viruses. Moreover, we found that targeting only the formation of replicase complexes did not stop the viral replication in vitro early in infection, while inhibiting intracellular trafficking processes may even lead to amplified viral growth. Author summary Plus-strand RNA viruses comprise a large group of related and medically relevant viruses. The current global pandemic of COVID-19 caused by the SARS-coronavirus-2 as well as the constant spread of diseases such as dengue and chikungunya fever show the necessity of a comprehensive and precise analysis of plus-strand RNA virus infections. Plus-strand RNA viruses share similarities in their life cycle. To understand their within-host replication strategies, we developed a mathematical model that studies pan-viral similarities and virus-specific differences of three plus-strand RNA viruses, namely hepatitis C, dengue, and coxsackievirus. By fitting our model to in vitro data, we found that only small virus-specific variations in the model were required to describe the dynamics of all three viruses. Furthermore, our model predicted that ribosomes involved in viral RNA translation seem to be a key player in plus-strand RNA replication efficiency, which may determine acute or chronic infection outcome. Furthermore, our in-silico drug treatment analysis suggests that targeting viral proteases involved in polyprotein cleavage, in combination with viral RNA replication, may represent promising drug targets with broad-spectrum antiviral activity.
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Affiliation(s)
- Carolin Zitzmann
- Institute of Bioinformatics, University Medicine Greifswald, Greifswald, Germany
- Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
| | - Christopher Dächert
- Research Group “Dynamics of Early Viral Infection and the Innate Antiviral Response”, Division Virus-Associated Carcinogenesis (F170), German Cancer Research Center (DKFZ), Heidelberg, Germany
| | - Bianca Schmid
- Dept of Infectious Diseases, Molecular Virology, Heidelberg University, Heidelberg, Germany
| | - Hilde van der Schaar
- Division of infectious Diseases and Immunology, Virology Section, Dept of Biomolecular Health Sciences, Utrecht University, Utrecht, The Netherlands
| | - Martijn van Hemert
- Department of Medical Microbiology, Leiden University Medical Center, Leiden, The Netherlands
| | - Alan S. Perelson
- Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
| | - Frank J.M. van Kuppeveld
- Division of infectious Diseases and Immunology, Virology Section, Dept of Biomolecular Health Sciences, Utrecht University, Utrecht, The Netherlands
| | - Ralf Bartenschlager
- Division Virus-Associated Carcinogenesis (F170), German Cancer Research Center (DKFZ), Heidelberg, Germany
- Dept of Infectious Diseases, Molecular Virology, Heidelberg University, Heidelberg, Germany
- German Center for Infection Research (DZIF), Heidelberg partner site, Heidelberg, Germany
| | - Marco Binder
- Research Group “Dynamics of Early Viral Infection and the Innate Antiviral Response”, Division Virus-Associated Carcinogenesis (F170), German Cancer Research Center (DKFZ), Heidelberg, Germany
| | - Lars Kaderali
- Institute of Bioinformatics, University Medicine Greifswald, Greifswald, Germany
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Fatehi F, Bingham RJ, Dechant PP, Stockley PG, Twarock R. Therapeutic interfering particles exploiting viral replication and assembly mechanisms show promising performance: a modelling study. Sci Rep 2021; 11:23847. [PMID: 34903795 PMCID: PMC8668974 DOI: 10.1038/s41598-021-03168-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/18/2021] [Accepted: 11/22/2021] [Indexed: 11/09/2022] Open
Abstract
Defective interfering particles arise spontaneously during a viral infection as mutants lacking essential parts of the viral genome. Their ability to replicate in the presence of the wild-type (WT) virus (at the expense of viable viral particles) is mimicked and exploited by therapeutic interfering particles. We propose a strategy for the design of therapeutic interfering RNAs (tiRNAs) against positive-sense single-stranded RNA viruses that assemble via packaging signal-mediated assembly. These tiRNAs contain both an optimised version of the virus assembly manual that is encoded by multiple dispersed RNA packaging signals and a replication signal for viral polymerase, but lack any protein coding information. We use an intracellular model for hepatitis C viral (HCV) infection that captures key aspects of the competition dynamics between tiRNAs and viral genomes for virally produced capsid protein and polymerase. We show that only a small increase in the assembly and replication efficiency of the tiRNAs compared with WT virus is required in order to achieve a treatment efficacy greater than 99%. This demonstrates that the proposed tiRNA design could be a promising treatment option for RNA viral infections.
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Affiliation(s)
- Farzad Fatehi
- York Cross-disciplinary Centre for Systems Analysis, University of York, York, YO10 5GE, UK
- Department of Mathematics, University of York, York, YO10 5DD, UK
| | - Richard J Bingham
- York Cross-disciplinary Centre for Systems Analysis, University of York, York, YO10 5GE, UK
- Department of Mathematics, University of York, York, YO10 5DD, UK
- Department of Biology, University of York, York, YO10 5DD, UK
| | - Pierre-Philippe Dechant
- York Cross-disciplinary Centre for Systems Analysis, University of York, York, YO10 5GE, UK
- Department of Mathematics, University of York, York, YO10 5DD, UK
- School of Science, Technology and Health, York St John University, York, YO31 7EX, UK
| | - Peter G Stockley
- Astbury Centre for Structural Molecular Biology, University of Leeds, Leeds, LS2 9JT, UK.
| | - Reidun Twarock
- York Cross-disciplinary Centre for Systems Analysis, University of York, York, YO10 5GE, UK.
- Department of Mathematics, University of York, York, YO10 5DD, UK.
- Department of Biology, University of York, York, YO10 5DD, UK.
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Churkin A, Reinharz V, Lewkiewicz S, Dahari H, Barash D. HCVMultiscaleFit: A Simulator For Parameter Estimation in Multiscale Models Of Hepatitis C Virus Dynamics. AIP CONFERENCE PROCEEDINGS 2020; 2293:420028. [PMID: 33349734 PMCID: PMC7750099 DOI: 10.1063/5.0026600] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
Abstract
Callibration in mathematical models that are based on differential equations is known to be of fundamental importance. For sophisticated models such as age-structured models that simulate biological agents, parameter estimation or fitting (callibration) that solves all cases of data points available presents a formidable challenge, as efficiency considerations need to be employed in order for the method to become practical. In the case of multiscale models of hepatitis C virus dynamics that deal with partial differential equations (PDEs), a fully numerical parameter estimation method was developed that does not require an analytical approximation of the solution to the multiscale model equations, avoiding the necessity to derive the long-term approximation for each model. However, the method is considerably slow because of precision problems in estimating derivatives with respect to the parameters near their boundary values, making it almost impractical for general use. In order to overcome this limitation, two steps have been taken that significantly reduce the running time by orders of magnitude and thereby lead to a practical method. First, constrained optimization is used, letting the user add constraints relating to the boundary values of each parameter before the method is executed. Second, optimization is performed by derivative-free methods, eliminating the need to evaluate expensive numerical derviative approximations. These steps that were successful in significantly speeding up a highly non-efficient approach, rendering it practical, can also be adapted to multiscale models of other viruses and other sophisticated differential equation models. The newly efficient methods that were developed as a result of the above approach are described. Illustrations are provided using a user-friendly simulator that incorporates the efficient methods for multiscale models. We provide a simulator called HCVMultiscaleFit with a Graphical User Interface that applies these methods and is useful to perform parameter estimation for simulating viral dynamics during antiviral treatment.
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Affiliation(s)
- Alexander Churkin
- Department of Software Engineering, Sami Shamoon College of Engineering, Beer-Sheva, Israel
| | - Vladimir Reinharz
- Center for Soft and Living Matter, Institute for Basic Science, Ulsan, South Korea
| | - Stephanie Lewkiewicz
- Department of Mathematics, University of California at Los Angeles, Los Angeles, California, USA
| | - Harel Dahari
- The Program for Experimental and Theoretical Modeling, Division of Hepatology, Department of Medicine, Stritch School of Medicine, Loyola University Chicago, Maywood, Illinois, USA
| | - Danny Barash
- Department of Computer Science, Ben-Gurion Universty, Beer-Sheva, Israel
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4
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Zitzmann C, Kaderali L, Perelson AS. Mathematical modeling of hepatitis C RNA replication, exosome secretion and virus release. PLoS Comput Biol 2020; 16:e1008421. [PMID: 33151933 PMCID: PMC7671504 DOI: 10.1371/journal.pcbi.1008421] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2020] [Revised: 11/17/2020] [Accepted: 10/06/2020] [Indexed: 01/04/2023] Open
Abstract
Hepatitis C virus (HCV) causes acute hepatitis C and can lead to life-threatening complications if it becomes chronic. The HCV genome is a single plus strand of RNA. Its intracellular replication is a spatiotemporally coordinated process of RNA translation upon cell infection, RNA synthesis within a replication compartment, and virus particle production. While HCV is mainly transmitted via mature infectious virus particles, it has also been suggested that HCV-infected cells can secrete HCV RNA carrying exosomes that can infect cells in a receptor independent manner. In order to gain insight into these two routes of transmission, we developed a series of intracellular HCV replication models that include HCV RNA secretion and/or virus assembly and release. Fitting our models to in vitro data, in which cells were infected with HCV, suggests that initially most secreted HCV RNA derives from intracellular cytosolic plus-strand RNA, but subsequently secreted HCV RNA derives equally from the cytoplasm and the replication compartments. Furthermore, our model fits to the data suggest that the rate of virus assembly and release is limited by host cell resources. Including the effects of direct acting antivirals in our models, we found that in spite of decreasing intracellular HCV RNA and extracellular virus concentration, low level HCV RNA secretion may continue as long as intracellular RNA is available. This may possibly explain the presence of detectable levels of plasma HCV RNA at the end of treatment even in patients that ultimately attain a sustained virologic response. Approximately 70 million people are chronically infected with hepatitis C virus (HCV), which if left untreated may lead to cirrhosis and liver cancer. However, modern drug therapy is highly effective and hepatitis C is the first chronic virus infection that can be cured with short-term therapy in almost all infected individuals. The within-host transmission of HCV occurs mainly via infectious virus particles, but experimental studies suggest that there may be additional receptor-independent cell-to-cell transmission by exosomes that carry the HCV genome. In order to understand the intracellular HCV lifecycle and HCV RNA spread, we developed a series of mathematical models that take both exosomal secretion and viral secretion into account. By fitting these models to in vitro data, we found that secretion of both HCV RNA as well as virus probably occurs and that the rate of virus assembly is likely limited by cellular co-factors on which the virus strongly depends for its own replication. Furthermore, our modeling predicted that the parameters governing the processes in the viral lifecycle that are targeted by direct acting antivirals are the most sensitive to perturbations, which may help explain their ability to cure this infection.
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Affiliation(s)
- Carolin Zitzmann
- University Medicine Greifswald, Institute of Bioinformatics and Center for Functional Genomics of Microbes, Greifswald, Germany
- Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
| | - Lars Kaderali
- University Medicine Greifswald, Institute of Bioinformatics and Center for Functional Genomics of Microbes, Greifswald, Germany
| | - Alan S. Perelson
- Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, New Mexico, United States of America
- * E-mail:
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Liao LE, Carruthers J, Smither SJ, Weller SA, Williamson D, Laws TR, García-Dorival I, Hiscox J, Holder BP, Beauchemin CAA, Perelson AS, López-García M, Lythe G, Barr JN, Molina-París C. Quantification of Ebola virus replication kinetics in vitro. PLoS Comput Biol 2020; 16:e1008375. [PMID: 33137116 PMCID: PMC7660928 DOI: 10.1371/journal.pcbi.1008375] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/23/2020] [Revised: 11/12/2020] [Accepted: 09/23/2020] [Indexed: 12/17/2022] Open
Abstract
Mathematical modelling has successfully been used to provide quantitative descriptions of many viral infections, but for the Ebola virus, which requires biosafety level 4 facilities for experimentation, modelling can play a crucial role. Ebola virus modelling efforts have primarily focused on in vivo virus kinetics, e.g., in animal models, to aid the development of antivirals and vaccines. But, thus far, these studies have not yielded a detailed specification of the infection cycle, which could provide a foundational description of the virus kinetics and thus a deeper understanding of their clinical manifestation. Here, we obtain a diverse experimental data set of the Ebola virus infection in vitro, and then make use of Bayesian inference methods to fully identify parameters in a mathematical model of the infection. Our results provide insights into the distribution of time an infected cell spends in the eclipse phase (the period between infection and the start of virus production), as well as the rate at which infectious virions lose infectivity. We suggest how these results can be used in future models to describe co-infection with defective interfering particles, which are an emerging alternative therapeutic.
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Affiliation(s)
- Laura E. Liao
- Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, NM, USA 87545
| | - Jonathan Carruthers
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
| | | | | | - Simon A. Weller
- Defence Science and Technology Laboratory, Salisbury SP4 0JQ, UK
| | - Diane Williamson
- Defence Science and Technology Laboratory, Salisbury SP4 0JQ, UK
| | - Thomas R. Laws
- Defence Science and Technology Laboratory, Salisbury SP4 0JQ, UK
| | - Isabel García-Dorival
- Institute of Infection and Global Health, University of Liverpool, Liverpool, L69 7BE, UK
| | - Julian Hiscox
- Institute of Infection and Global Health, University of Liverpool, Liverpool, L69 7BE, UK
| | - Benjamin P. Holder
- Department of Physics, Grand Valley State University, Allendale, MI, USA 49401
| | - Catherine A. A. Beauchemin
- Department of Physics, Ryerson University, Toronto, ON, Canada M5B 2K3
- Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS) Research Program at RIKEN, Wako, Saitama, Japan, 351-0198
| | - Alan S. Perelson
- Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, NM, USA 87545
| | - Martín López-García
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
| | - Grant Lythe
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
| | - John N. Barr
- School of Molecular and Cellular Biology, University of Leeds, Leeds LS2 9JT, UK
| | - Carmen Molina-París
- Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
- * E-mail:
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6
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Churkin A, Lewkiewicz S, Reinharz V, Dahari H, Barash D. Efficient Methods for Parameter Estimation of Ordinary and Partial Differential Equation Models of Viral Hepatitis Kinetics. MATHEMATICS 2020; 8. [PMID: 33224865 PMCID: PMC7676746 DOI: 10.3390/math8091483] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
Parameter estimation in mathematical models that are based on differential equations is known to be of fundamental importance. For sophisticated models such as age-structured models that simulate biological agents, parameter estimation that addresses all cases of data points available presents a formidable challenge and efficiency considerations need to be employed in order for the method to become practical. In the case of age-structured models of viral hepatitis dynamics under antiviral treatment that deal with partial differential equations, a fully numerical parameter estimation method was developed that does not require an analytical approximation of the solution to the multiscale model equations, avoiding the necessity to derive the long-term approximation for each model. However, the method is considerably slow because of precision problems in estimating derivatives with respect to the parameters near their boundary values, making it almost impractical for general use. In order to overcome this limitation, two steps have been taken that significantly reduce the running time by orders of magnitude and thereby lead to a practical method. First, constrained optimization is used, letting the user add constraints relating to the boundary values of each parameter before the method is executed. Second, optimization is performed by derivative-free methods, eliminating the need to evaluate expensive numerical derivative approximations. The newly efficient methods that were developed as a result of the above approach are described for hepatitis C virus kinetic models during antiviral therapy. Illustrations are provided using a user-friendly simulator that incorporates the efficient methods for both the ordinary and partial differential equation models.
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Affiliation(s)
- Alexander Churkin
- Department of Software Engineering, Sami Shamoon College of Engineering, Beer-Sheva 8410501, Israel
- Correspondence: (A.C.); (D.B.); Tel.: +972-8-647-5281 (A.C.); +972-8-647-2714 (D.B.)
| | - Stephanie Lewkiewicz
- Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095, USA
| | - Vladimir Reinharz
- Department of Computer Science, Université du Québec à Montréal, Montreal, QC H3C 3P8, Canada
| | - Harel Dahari
- Program for Experimental and Theoretical Modeling, Division of Hepatology, Department of Medicine, Stritch School of Medicine, Loyola University Medical Center, Maywoood, IL 60153, USA
| | - Danny Barash
- Department of Computer Science, Ben-Gurion University, Beer-Sheva 8410501, Israel
- Correspondence: (A.C.); (D.B.); Tel.: +972-8-647-5281 (A.C.); +972-8-647-2714 (D.B.)
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7
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Reis RF, de Melo Quintela B, de Oliveira Campos J, Gomes JM, Rocha BM, Lobosco M, Weber dos Santos R. Characterization of the COVID-19 pandemic and the impact of uncertainties, mitigation strategies, and underreporting of cases in South Korea, Italy, and Brazil. CHAOS, SOLITONS, AND FRACTALS 2020; 136:109888. [PMID: 32412556 PMCID: PMC7221372 DOI: 10.1016/j.chaos.2020.109888] [Citation(s) in RCA: 54] [Impact Index Per Article: 13.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/11/2020] [Revised: 05/06/2020] [Accepted: 05/10/2020] [Indexed: 05/09/2023]
Abstract
By April 7th, 2020, the Coronavirus disease 2019 (COVID-19) has infected one and a half million people worldwide, accounting for over 80 thousand of deaths in 209 countries and territories around the world. The new and fast dynamics of the pandemic are challenging the health systems of different countries. In the absence of vaccines or effective treatments, mitigation policies, such as social isolation and lock-down of cities, have been adopted, but the results vary among different countries. Some countries were able to control the disease at the moment, as is the case of South Korea. Others, like Italy, are now experiencing the peak of the pandemic. Finally, countries with emerging economies and social issues, like Brazil, are in the initial phase of the pandemic. In this work, we use mathematical models with time-dependent coefficients, techniques of inverse and forward uncertainty quantification, and sensitivity analysis to characterize essential aspects of the COVID-19 in the three countries mentioned above. The model parameters estimated for South Korea revealed effective social distancing and isolation policies, border control, and a high number in the percentage of reported cases. In contrast, underreporting of cases was estimated to be very high in Brazil and Italy. In addition, the model estimated a poor isolation policy at the moment in Brazil, with a reduction of contact around 40%, whereas Italy and South Korea estimated numbers for contact reduction are at 75% and 90%, respectively. This characterization of the COVID-19, in these different countries under different scenarios and phases of the pandemic, supports the importance of mitigation policies, such as social distancing. In addition, it raises serious concerns for socially and economically fragile countries, where underreporting poses additional challenges to the management of the COVID-19 pandemic by significantly increasing the uncertainties regarding its dynamics.
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Affiliation(s)
- Ruy Freitas Reis
- Departamento de Ciência da Computação, Universidade Federal de Juiz de Fora, Brazil
| | - Bárbara de Melo Quintela
- Departamento de Ciência da Computação, Universidade Federal de Juiz de Fora, Brazil
- Department of Industrial Engineering, Alma Mater Studiorum - University of Bologna, Italy
| | | | - Johnny Moreira Gomes
- Pós-Graduação em Modelagem Computacional, Universidade Federal de Juiz de Fora, Brazil
| | - Bernardo Martins Rocha
- Departamento de Ciência da Computação, Universidade Federal de Juiz de Fora, Brazil
- Pós-Graduação em Modelagem Computacional, Universidade Federal de Juiz de Fora, Brazil
| | - Marcelo Lobosco
- Departamento de Ciência da Computação, Universidade Federal de Juiz de Fora, Brazil
- Pós-Graduação em Modelagem Computacional, Universidade Federal de Juiz de Fora, Brazil
| | - Rodrigo Weber dos Santos
- Departamento de Ciência da Computação, Universidade Federal de Juiz de Fora, Brazil
- Pós-Graduação em Modelagem Computacional, Universidade Federal de Juiz de Fora, Brazil
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8
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Garira W. The research and development process for multiscale models of infectious disease systems. PLoS Comput Biol 2020; 16:e1007734. [PMID: 32240165 PMCID: PMC7156109 DOI: 10.1371/journal.pcbi.1007734] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/15/2019] [Revised: 04/14/2020] [Accepted: 02/13/2020] [Indexed: 01/26/2023] Open
Abstract
Multiscale modelling of infectious disease systems falls within the domain of complexity science—the study of complex systems. However, what should be made clear is that current progress in multiscale modelling of infectious disease dynamics is still as yet insufficient to present it as a mature sub-discipline of complexity science. In this article we present a methodology for development of multiscale models of infectious disease systems. This methodology is a set of partially ordered research and development activities that result in multiscale models of infectious disease systems built from different scientific approaches. Therefore, the conclusive result of this article is a methodology to design multiscale models of infectious diseases. Although this research and development process for multiscale models cannot be claimed to be unique and final, it constitutes a good starting point, which may be found useful as a basis for further refinement in the discourse for multiscale modelling of infectious disease dynamics. Complex systems such as infectious disease systems are inherently multilevel and multiscale systems. The study of such complex systems is called complexity science. In this article we present a methodology to design multiscale models of infectious disease systems from a complex systems perspective. Within this perspective we define complexity science as the study of the interconnected relationships of the levels and scales of organization of a complex system. We therefore, define the degree of complexity of a complex system as the number of levels and scales of organization of the complex system needed to describe it. In this work we first present a common multiscale vision of the multilevel and multiscale structure of infectious disease systems as complex systems in which the levels and scales of organization of an infectious disease system interact through different self-sustained multiscale cycles/loops (primary multiscale loops, or secondary multiscale loops, or tertiary multiscale loops) formed at different levels of organization of an infectious disease system due to ongoing reciprocal influence between the microscale and the macroscale. Guided by this multiscale vision, we propose a four-stage research and development process that result in multiscale models of infectious disease systems built from different scientific approaches.
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Affiliation(s)
- Winston Garira
- Modelling Health and Environmental Linkages Research Group (MHELRG), Department of Mathematics and Applied Mathematics, University of Venda, Thohoyandou, South Africa
- * E-mail: ,
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9
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Reinharz V, Churkin A, Lewkiewicz S, Dahari H, Barash D. A Parameter Estimation Method for Multiscale Models of Hepatitis C Virus Dynamics. Bull Math Biol 2019; 81:3675-3721. [PMID: 31338739 PMCID: PMC7375976 DOI: 10.1007/s11538-019-00644-7] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2018] [Accepted: 07/10/2019] [Indexed: 12/11/2022]
Abstract
Mathematical models that are based on differential equations require detailed knowledge about the parameters that are included in the equations. Some of the parameters can be measured experimentally while others need to be estimated. When the models become more sophisticated, such as in the case of multiscale models of hepatitis C virus dynamics that deal with partial differential equations (PDEs), several strategies can be tried. It is possible to use parameter estimation on an analytical approximation of the solution to the multiscale model equations, namely the long-term approximation, but this limits the scope of the parameter estimation method used and a long-term approximation needs to be derived for each model. It is possible to transform the PDE multiscale model to a system of ODEs, but this has an effect on the model parameters themselves and the transformation can become problematic for some models. Finally, it is possible to use numerical solutions for the multiscale model and then use canned methods for the parameter estimation, but the latter is making the user dependent on a black box without having full control over the method. The strategy developed here is to start by working directly on the multiscale model equations for preparing them toward the parameter estimation method that is fully coded and controlled by the user. It can also be adapted to multiscale models of other viruses. The new method is described, and illustrations are provided using a user-friendly simulator that incorporates the method.
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Affiliation(s)
- Vladimir Reinharz
- Department of Computer Science, Ben-Gurion University, Beersheba, Israel
| | - Alexander Churkin
- Department of Software Engineering, Sami Shamoon College of Engineering, Beersheba, Israel
| | - Stephanie Lewkiewicz
- Department of Mathematics, University of California at Los Angeles, Los Angeles, CA, USA
| | - Harel Dahari
- Program for Experimental and Theoretical Modeling, Division of Hepatology, Department of Medicine, Loyola University Medical Center, Maywoood, IL, USA
| | - Danny Barash
- Department of Computer Science, Ben-Gurion University, Beersheba, Israel.
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Knodel MM, Targett-Adams P, Grillo A, Herrmann E, Wittum G. Advanced Hepatitis C Virus Replication PDE Models within a Realistic Intracellular Geometric Environment. INTERNATIONAL JOURNAL OF ENVIRONMENTAL RESEARCH AND PUBLIC HEALTH 2019; 16:E513. [PMID: 30759770 PMCID: PMC6388173 DOI: 10.3390/ijerph16030513] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/29/2018] [Revised: 01/08/2019] [Accepted: 01/16/2019] [Indexed: 02/06/2023]
Abstract
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.
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Affiliation(s)
- Markus M Knodel
- Department of Mathematics, Chair of Applied Mathematics 1, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany.
| | | | - Alfio Grillo
- Dipartimento di Scienze Matematiche (DISMA) "G.L. Lagrange", Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino (TO), Italy.
| | - Eva Herrmann
- Department of Medicine, Institute for Biostatistics and Mathematic Modeling, Goethe Universität Frankfurt, Theodor-Stern-Kai 7, 60590 Frankfurt am Main, Germany.
| | - Gabriel Wittum
- Goethe Center for Scientific Computing (G-CSC), Goethe Universität Frankfurt, Kettenhofweg 139, 60325 Frankfurt am Main, Germany.
- Applied Mathematics and Computational Science, King Abdullah University of Science and Technology (KAUST), 23955-6900 Thuwal, Saudi Arabia.
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Mathematical Analysis of a Transformed ODE from a PDE Multiscale Model of Hepatitis C Virus Infection. Bull Math Biol 2019; 81:1427-1441. [PMID: 30644067 DOI: 10.1007/s11538-018-00564-y] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2018] [Accepted: 12/27/2018] [Indexed: 02/06/2023]
Abstract
Mathematical modeling has revealed the quantitative dynamics during the process of viral infection and evolved into an important tool in modern virology. Coupled with analyses of clinical and experimental data, the widely used basic model of viral dynamics described by ordinary differential equations (ODEs) has been well parameterized. In recent years, age-structured models, called "multiscale model," formulated by partial differential equations (PDEs) have also been developed and become useful tools for more detailed data analysis. However, in general, PDE models are considerably more difficult to subject to mathematical and numerical analyses. In our recently reported study, we successfully derived a mathematically identical ODE model from a PDE model, which helps to overcome the limitations of the PDE model with regard to clinical data analysis. Here, we derive the basic reproduction number from the identical ODE model and provide insight into the global stability of all possible steady states of the ODE model.
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