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Ruth B, Peter S, Ibrahim B, Dittrich P. Revealing the hierarchical structure of microbial communities. Sci Rep 2024; 14:11202. [PMID: 38755262 PMCID: PMC11099180 DOI: 10.1038/s41598-024-61836-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/28/2023] [Accepted: 05/10/2024] [Indexed: 05/18/2024] Open
Abstract
Measuring the dynamics of microbial communities results in high-dimensional measurements of taxa abundances over time and space, which is difficult to analyze due to complex changes in taxonomic compositions. This paper presents a new method to investigate and visualize the intrinsic hierarchical community structure implied by the measurements. The basic idea is to identify significant intersection sets, which can be seen as sub-communities making up the measured communities. Using the subset relationship, the intersection sets together with the measurements form a hierarchical structure visualized as a Hasse diagram. Chemical organization theory (COT) is used to relate the hierarchy of the sets of taxa to potential taxa interactions and to their potential dynamical persistence. The approach is demonstrated on a data set of community data obtained from bacterial 16S rRNA gene sequencing for samples collected monthly from four groundwater wells over a nearly 3-year period (n = 114) along a hillslope area. The significance of the hierarchies derived from the data is evaluated by showing that they significantly deviate from a random model. Furthermore, it is demonstrated how the hierarchy is related to temporal and spatial factors; and how the idea of a core microbiome can be extended to a set of interrelated core microbiomes. Together the results suggest that the approach can support developing models of taxa interactions in the future.
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Affiliation(s)
- Beatrice Ruth
- Department of Mathematics and Computer Science, Friedrich Schiller University Jena, Fürstengraben, 07743, Jena, Germany
| | - Stephan Peter
- Department of Basic Sciences, Ernst-Abbe University of Applied Sciences Jena, Carl-Zeiss-Promenade 2, 07745, Jena, Germany
| | - Bashar Ibrahim
- Department of Mathematics and Computer Science, Friedrich Schiller University Jena, Fürstengraben, 07743, Jena, Germany.
- Department of Mathematics & Natural Sciences and Centre for Applied Mathematics & Bioinformatics, Gulf University for Science and Technology, 32093, Hawally, Kuwait.
- European Virus Bioinformatics Center, Leutragraben 1, 07743, Jena, Germany.
| | - Peter Dittrich
- Department of Mathematics and Computer Science, Friedrich Schiller University Jena, Fürstengraben, 07743, Jena, Germany.
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Cai Y, Zhao Z, Zhuge C. The spatial dynamics of immune response upon virus infection through hybrid dynamical computational model. Front Immunol 2023; 14:1257953. [PMID: 38022575 PMCID: PMC10657891 DOI: 10.3389/fimmu.2023.1257953] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/13/2023] [Accepted: 10/10/2023] [Indexed: 12/01/2023] Open
Abstract
Introduction The immune responses play important roles in the course of disease initiation and progression upon virus infection such as SARS-CoV-2. As the tissues consist of spatial structures, the spatial dynamics of immune responses upon viral infection are essential to the outcome of infection. Methods A hybrid computational model based on cellular automata coupled with partial differential equations is developed to simulate the spatial patterns and dynamics of the immune responses of tissue upon virus infection with several different immune movement modes. Results Various patterns of the distribution of virus particles under different immune strengths and movement modes of immune cells are obtained through the computational models. The results also reveal that the directed immune cell wandering model has a better immunization effect. Several other characteristics, such as the peak level of virus density and onset time and the onset of the diseases, are also checked with different immune and physiological conditions, for example, different immune clearance strengths, and different cell-to-cell transmission rates. Furthermore, by the Lasso analysis, it is identified that the three main parameters had the most impact on the rate of onset time of disease. It is also shown that the cell-to-cell transmission rate has a significant effect and is more important for controlling the diseases than those for the cell-free virus given that the faster cell-to-cell transmission than cell-free transmission the rate of virus release is low. Discussion Our model simulates the process of viral and immune response interactions in the alveola repithelial tissues of infected individuals, providing insights into the viral propagation of viruses in two dimensions as well as the influence of immune response patterns and key factors on the course of infection.
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Affiliation(s)
| | | | - Changjing Zhuge
- Department of Mathematics, Faculty of Science, Beijing University of Technology, Beijing, China
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Peter S, Woitke L, Dittrich P, Ibrahim B. Computing all persistent subspaces of a reaction-diffusion system. Sci Rep 2023; 13:17169. [PMID: 37821664 PMCID: PMC10567720 DOI: 10.1038/s41598-023-44244-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/05/2023] [Accepted: 10/05/2023] [Indexed: 10/13/2023] Open
Abstract
An algorithm is presented for computing a reaction-diffusion partial differential equation (PDE) system for all possible subspaces that can hold a persistent solution of the equation. For this, all possible sub-networks of the underlying reaction network that are distributed organizations (DOs) are identified. Recently it has been shown that a persistent subspace must be a DO. The algorithm computes the hierarchy of DOs starting from the largest by a linear programming approach using integer cuts. The underlying constraints use elementary reaction closures as minimal building blocks to guarantee local closedness and global self-maintenance, required for a DO. Additionally, the algorithm delivers for each subspace an affiliated set of organizational reactions and minimal compartmentalization that is necessary for this subspace to persist. It is proved that all sets of organizational reactions of a reaction network, as already DOs, form a lattice. This lattice contains all potentially persistent sets of reactions of all constrained solutions of reaction-diffusion PDEs. This provides a hierarchical structure of all persistent subspaces with regard to the species and also to the reactions of the reaction-diffusion PDE system. Here, the algorithm is described and the corresponding Python source code is provided. Furthermore, an analysis of its run time is performed and all models from the BioModels database as well as further examples are examined. Apart from the practical implications of the algorithm the results also give insights into the complexity of solving reaction-diffusion PDEs.
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Affiliation(s)
- Stephan Peter
- Department of Basic Sciences, Ernst-Abbe University of Applied Sciences Jena, Carl-Zeiss-Promenade 2, 07745, Jena, Germany
| | - Linus Woitke
- Department of Mathematics and Computer Science, Friedrich Schiller University Jena, Fürstengraben, 07743, Jena, Germany
| | - Peter Dittrich
- Department of Mathematics and Computer Science, Friedrich Schiller University Jena, Fürstengraben, 07743, Jena, Germany.
| | - Bashar Ibrahim
- Department of Mathematics and Computer Science, Friedrich Schiller University Jena, Fürstengraben, 07743, Jena, Germany.
- Department of Mathematics & Natural Sciences and Centre for Applied Mathematics & Bioinformatics, Gulf University for Science and Technology, 32093, Hawally, Kuwait.
- European Virus Bioinformatics Center, Leutragraben 1, 07743, Jena, Germany.
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Infectious Disease Modeling with Socio-Viral Behavioral Aspects-Lessons Learned from the Spread of SARS-CoV-2 in a University. Trop Med Infect Dis 2022; 7:tropicalmed7100289. [PMID: 36288030 PMCID: PMC9608982 DOI: 10.3390/tropicalmed7100289] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2022] [Revised: 10/04/2022] [Accepted: 10/05/2022] [Indexed: 11/04/2022] Open
Abstract
When it comes to understanding the spread of COVID-19, recent studies have shown that pathogens can be transmitted in two ways: direct contact and airborne pathogens. While the former is strongly related to the distancing behavior of people in society, the latter are associated with the length of the period in which the airborne pathogens remain active. Considering those facts, we constructed a compartmental model with a time-dependent transmission rate that incorporates the two sources of infection. This paper provides an analytical and numerical study of the model that validates trivial insights related to disease spread in a responsive society. As a case study, we applied the model to the COVID-19 spread data from a university environment, namely, the Institut Teknologi Bandung, Indonesia, during its early reopening stage, with a constant number of students. The results show a significant fit between the rendered model and the recorded cases of infections. The extrapolated trajectories indicate the resurgence of cases as students' interaction distance approaches its natural level. The assessment of several strategies is undertaken in this study in order to assist with the school reopening process.
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Wang Y, Wang P, Zhang S, Pan H. Uncertainty Modeling of a Modified SEIR Epidemic Model for COVID-19. BIOLOGY 2022; 11:biology11081157. [PMID: 36009784 PMCID: PMC9404969 DOI: 10.3390/biology11081157] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/29/2022] [Revised: 07/27/2022] [Accepted: 07/30/2022] [Indexed: 06/01/2023]
Abstract
Based on SEIR (susceptible-exposed-infectious-removed) epidemic model, we propose a modified epidemic mathematical model to describe the spread of the coronavirus disease 2019 (COVID-19) epidemic in Wuhan, China. Using public data, the uncertainty parameters of the proposed model for COVID-19 in Wuhan were calibrated. The uncertainty of the control basic reproduction number was studied with the posterior probability density function of the uncertainty model parameters. The mathematical model was used to inverse deduce the earliest start date of COVID-19 infection in Wuhan with consideration of the lack of information for the initial conditions of the model. The result of the uncertainty analysis of the model is in line with the observed data for COVID-19 in Wuhan, China. The numerical results show that the modified mathematical model could model the spread of COVID-19 epidemics.
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Frank T. SARS-coronavirus-2 infections: biological instabilities characterized by order parameters. Phys Biol 2022; 19. [PMID: 35108687 DOI: 10.1088/1478-3975/ac5155] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2021] [Accepted: 02/02/2022] [Indexed: 11/12/2022]
Abstract
A four-variable virus dynamics TIIV model was considered that involves infected cells in an eclipse phase. The state space description of the model was transferred into an amplitude space description which is the appropriate general, nonlinear physics framework to describe instabilities. In this context, the unstable eigenvector or order parameter of the model was determined. Subsequently, a model-based analysis of viral load data from eight symptomatic COVID-19 patients was conducted. For all patients, it was found that the initial SARS-CoV-2 infection evolved along the respective patient-specific order parameter, as expected by theoretical considerations. The order parameter amplitude that described the initial virus multiplication showed doubling times between 30 minutes and 3 hours. Peak viral loads of patients were linearly related to the amplitudes of the patient order parameters. Finally, it was found that the patient order parameters determined qualitatively and quantitatively the relationships between the increases in virus-producing infected cells and infected cells in the eclipse phase. Overall, the study echoes the 40 years old suggestion by Mackey and Glass to consider diseases as instabilities.
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Affiliation(s)
- Till Frank
- University of Connecticut, 406 Babbidge Road, Storrs, Connecticut, 06269, UNITED STATES
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Marzban S, Han R, Juhász N, Röst G. A hybrid PDE-ABM model for viral dynamics with application to SARS-CoV-2 and influenza. ROYAL SOCIETY OPEN SCIENCE 2021; 8:210787. [PMID: 34737874 PMCID: PMC8564626 DOI: 10.1098/rsos.210787] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/14/2021] [Accepted: 10/11/2021] [Indexed: 05/08/2023]
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.
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Affiliation(s)
- Sadegh Marzban
- Bolyai Institute, University of Szeged, Szeged 6720, Hungary
| | - Renji Han
- Bolyai Institute, University of Szeged, Szeged 6720, Hungary
| | - Nóra Juhász
- Bolyai Institute, University of Szeged, Szeged 6720, Hungary
| | - Gergely Röst
- Bolyai Institute, University of Szeged, Szeged 6720, Hungary
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Abstract
Coronavirus disease 2019 (COVID-19) is a respiratory disease caused by SARS-CoV-2. It appeared in China in late 2019 and rapidly spread to most countries of the world. Cancer patients infected with SARS-CoV-2 are at higher risk of developing severe infection and death. This risk increases further in the presence of lymphopenia affecting the lymphocytes count. Here, we develop a delayed within-host SARS-CoV-2/cancer model. The model describes the occurrence of SARS-CoV-2 infection in cancer patients and its effect on the functionality of immune responses. The model considers the time delays that affect the growth rates of healthy epithelial cells and cancer cells. We provide a detailed analysis of the model by proving the nonnegativity and boundedness of the solutions, finding steady states, and showing the global stability of the different steady states. We perform numerical simulations to highlight some important observations. The results indicate that increasing the time delay in the growth rate of cancer cells reduced the size of tumors and decreased the likelihood of deterioration in the condition of SARS-CoV-2/cancer patients. On the other hand, lymphopenia increased the concentrations of SARS-CoV-2 particles and cancer cells, which worsened the condition of the patient.
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Hwang W, Lei W, Katritsis NM, MacMahon M, Chapman K, Han N. Current and prospective computational approaches and challenges for developing COVID-19 vaccines. Adv Drug Deliv Rev 2021; 172:249-274. [PMID: 33561453 PMCID: PMC7871111 DOI: 10.1016/j.addr.2021.02.004] [Citation(s) in RCA: 18] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2020] [Revised: 02/01/2021] [Accepted: 02/03/2021] [Indexed: 12/23/2022]
Abstract
SARS-CoV-2, which causes COVID-19, was first identified in humans in late 2019 and is a coronavirus which is zoonotic in origin. As it spread around the world there has been an unprecedented effort in developing effective vaccines. Computational methods can be used to speed up the long and costly process of vaccine development. Antigen selection, epitope prediction, and toxicity and allergenicity prediction are areas in which computational tools have already been applied as part of reverse vaccinology for SARS-CoV-2 vaccine development. However, there is potential for computational methods to assist further. We review approaches which have been used and highlight additional bioinformatic approaches and PK modelling as in silico methods which may be useful for SARS-CoV-2 vaccine design but remain currently unexplored. As more novel viruses with pandemic potential are expected to arise in future, these techniques are not limited to application to SARS-CoV-2 but also useful to rapidly respond to novel emerging viruses.
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Affiliation(s)
- Woochang Hwang
- Milner Therapeutics Institute, University of Cambridge, Cambridge, UK
| | - Winnie Lei
- Milner Therapeutics Institute, University of Cambridge, Cambridge, UK; Department of Surgery, University of Cambridge, Cambridge, UK
| | - Nicholas M Katritsis
- Milner Therapeutics Institute, University of Cambridge, Cambridge, UK; Department of Chemical Engineering and Biotechnology, University of Cambridge, Cambridge, UK
| | - Méabh MacMahon
- Milner Therapeutics Institute, University of Cambridge, Cambridge, UK; Centre for Therapeutics Discovery, LifeArc, Stevenage, UK
| | - Kathryn Chapman
- Milner Therapeutics Institute, University of Cambridge, Cambridge, UK
| | - Namshik Han
- Milner Therapeutics Institute, University of Cambridge, Cambridge, UK.
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Mathematical modeling of ventilator-induced lung inflammation. J Theor Biol 2021; 526:110738. [PMID: 33930440 DOI: 10.1016/j.jtbi.2021.110738] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2020] [Revised: 04/20/2021] [Accepted: 04/22/2021] [Indexed: 12/13/2022]
Abstract
Despite the benefits of mechanical ventilators, prolonged or misuse of ventilators may lead to ventilation-associated/ventilation-induced lung injury (VILI). Lung insults, such as respiratory infections and lung injuries, can damage the pulmonary epithelium, with the most severe cases needing mechanical ventilation for effective breathing and survival. Damaged epithelial cells within the alveoli trigger a local immune response. A key immune cell is the macrophage, which can differentiate into a spectrum of phenotypes ranging from pro- to anti-inflammatory. To gain a greater understanding of the mechanisms of the immune response to VILI and post-ventilation outcomes, we developed a mathematical model of interactions between the immune system and site of damage while accounting for macrophage phenotype. Through Latin hypercube sampling we generated a collection of parameter sets that are associated with a numerical steady state. We then simulated ventilation-induced damage using these steady state values as the initial conditions in order to evaluate how baseline immune state and lung health affect outcomes. We used a variety of methods to analyze the resulting parameter sets, transients, and outcomes, including a random forest decision tree algorithm and parameter sensitivity with eFAST. Analysis shows that parameters and properties of transients related to epithelial repair and M1 activation are important factors. Using the results of this analysis, we hypothesized interventions and used these treatment strategies to modulate the response to ventilation for particular parameters sets.
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Competitive exclusion during co-infection as a strategy to prevent the spread of a virus: A computational perspective. PLoS One 2021; 16:e0247200. [PMID: 33626106 PMCID: PMC7904198 DOI: 10.1371/journal.pone.0247200] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2020] [Accepted: 02/02/2021] [Indexed: 01/24/2023] Open
Abstract
Inspired by the competition exclusion principle, this work aims at providing a computational framework to explore the theoretical feasibility of viral co-infection as a possible strategy to reduce the spread of a fatal strain in a population. We propose a stochastic-based model—called Co-Wish—to understand how competition between two viruses over a shared niche can affect the spread of each virus in infected tissue. To demonstrate the co-infection of two viruses, we first simulate the characteristics of two virus growth processes separately. Then, we examine their interactions until one can dominate the other. We use Co-Wish to explore how the model varies as the parameters of each virus growth process change when two viruses infect the host simultaneously. We will also investigate the effect of the delayed initiation of each infection. Moreover, Co-Wish not only examines the co-infection at the cell level but also includes the innate immune response during viral infection. The results highlight that the waiting times in the five stages of the viral infection of a cell in the model—namely attachment, penetration, eclipse, replication, and release—play an essential role in the competition between the two viruses. While it could prove challenging to fully understand the therapeutic potentials of viral co-infection, we discuss that our theoretical framework hints at an intriguing research direction in applying co-infection dynamics in controlling any viral outbreak’s speed.
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