1
|
Adewole MO, Faniran TS, Abdullah FA, Ali MKM. COVID-19 dynamics and immune response: Linking within-host and between-host dynamics. Chaos Solitons Fractals 2023; 173:113722. [PMID: 38620099 PMCID: PMC10291298 DOI: 10.1016/j.chaos.2023.113722] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/02/2023] [Revised: 04/26/2023] [Accepted: 06/13/2023] [Indexed: 11/04/2023]
Abstract
The global impact of COVID-19 has led to the development of numerous mathematical models to understand and control the pandemic. However, these models have not fully captured how the disease's dynamics are influenced by both within-host and between-host factors. To address this, a new mathematical model is proposed that links these dynamics and incorporates immune response. The model is compartmentalized with a fractional derivative in the sense of Caputo-Fabrizio, and its properties are studied to show a unique solution. Parameter estimation is carried out by fitting real-life data, and sensitivity analysis is conducted using various methods. The model is then numerically implemented to demonstrate how the dynamics within infected hosts drive human-to-human transmission, and various intervention strategies are compared based on the percentage of averted deaths. The simulations suggest that a combination of medication to boost the immune system, prevent infected cells from producing the virus, and adherence to COVID-19 protocols is necessary to control the spread of the virus since no single intervention strategy is sufficient.
Collapse
Affiliation(s)
- Matthew O Adewole
- School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia
- Department of Computer Science and Mathematics, Mountain Top University, Prayer City, Ogun State, Nigeria
| | - Taye Samuel Faniran
- Laboratory de Mathematiques de Besancon, University of Franche-Comte, France
- Department of Computer Science, Lead City University, Ibadan, Nigeria
| | - Farah A Abdullah
- School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia
| | - Majid K M Ali
- School of Mathematical Sciences, Universiti Sains Malaysia, Malaysia
| |
Collapse
|
2
|
Gedeon T, Humphries AR, Mackey MC, Walther HO, Wang Z. Operon dynamics with state dependent transcription and/or translation delays. J Math Biol 2021; 84:2. [PMID: 34905089 DOI: 10.1007/s00285-021-01693-0] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2021] [Revised: 06/18/2021] [Accepted: 11/16/2021] [Indexed: 11/29/2022]
Abstract
Transcription and translation retrieve and operationalize gene encoded information in cells. These processes are not instantaneous and incur significant delays. In this paper we study Goodwin models of both inducible and repressible operons with state-dependent delays. The paper provides justification and derivation of the model, detailed analysis of the appropriate setting of the corresponding dynamical system, and extensive numerical analysis of its dynamics. Comparison with constant delay models shows significant differences in dynamics that include existence of stable periodic orbits in inducible systems and multistability in repressible systems. A combination of parameter space exploration, numerics, analysis of steady state linearization and bifurcation theory indicates the likely presence of Shilnikov-type homoclinic bifurcations in the repressible operon model.
Collapse
Affiliation(s)
- Tomáš Gedeon
- Department of Mathematics, Montana State University, Bozeman, MT, 59717, USA
| | - Antony R Humphries
- Departments of Mathematics and Statistics, and, Physiology, McGill University, Montreal, QC, H3A 0B9, Canada
| | - Michael C Mackey
- Departments of Physiology, Physics, and, Mathematics and Statistics, McGill University, 3655 Promenade Sir William Osler, Montreal, QC, H3G 1Y6, Canada
| | - Hans-Otto Walther
- Mathematisches Institut, Universität Giessen, Arndtstrasse 2, 35392, Giessen, Germany
| | - Zhao Wang
- Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 0B9, Canada.
| |
Collapse
|
3
|
Abstract
Cell division is a process that involves many biochemical steps and complex biophysical mechanisms. To simplify the understanding of what triggers cell division, three basic models that subsume more microscopic cellular processes associated with cell division have been proposed. Cells can divide based on the time elapsed since their birth, their size, and/or the volume added since their birth-the timer, sizer, and adder models, respectively. Here, we propose unified adder-sizer models and investigate some of the properties of different adder processes arising in cellular proliferation. Although the adder-sizer model provides a direct way to model cell population structure, we illustrate how it is mathematically related to the well-known model in which cell division depends on age and size. Existence and uniqueness of weak solutions to our 2+1-dimensional PDE model are proved, leading to the convergence of the discretized numerical solutions and allowing us to numerically compute the dynamics of cell population densities. We then generalize our PDE model to incorporate recent experimental findings of a system exhibiting mother-daughter correlations in cellular growth rates. Numerical experiments illustrating possible average cell volume blowup and the dynamical behavior of cell populations with mother-daughter correlated growth rates are carried out. Finally, motivated by new experimental findings, we extend our adder model cases where the controlling variable is the added size between DNA replication initiation points in the cell cycle.
Collapse
Affiliation(s)
- Mingtao Xia
- Department of Mathematics, UCLA, Los Angeles, CA 90095-1555
| | - Chris D Greenman
- School of Computing Sciences, University of East Anglia, Norwich, UK NR4 7TJ
| | - Tom Chou
- Departments of Computational Medicine and Mathematics, UCLA, Los Angeles, CA 90095-1766
| |
Collapse
|
4
|
Wang X, Tang S, Song X, Rong L. Mathematical analysis of an HIV latent infection model including both virus-to-cell infection and cell-to-cell transmission. J Biol Dyn 2017; 11:455-483. [PMID: 27730851 DOI: 10.1080/17513758.2016.1242784] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
HIV can infect cells via virus-to-cell infection or cell-to-cell viral transmission. These two infection modes may occur in a synergistic way and facilitate viral spread within an infected individual. In this paper, we developed an HIV latent infection model including both modes of transmission and time delays between viral entry and integration or viral production. We analysed the model by defining the basic reproductive number, showing the existence, positivity and boundedness of the solution, and proving the local and global stability of the infection-free and infected steady states. Numerical simulations have been performed to illustrate the theoretical results and evaluate the effects of time delays and fractions of infection leading to latency on the virus dynamics. The estimates of the relative contributions to the HIV latent reservoir and the virus population from the two modes of transmission have also been provided.
Collapse
Affiliation(s)
- Xia Wang
- a College of Mathematics and Information Science , Xinyang Normal University , Xinyang , People's Republic of China
| | - Sanyi Tang
- b College of Mathematics and Information Science , Shaanxi Normal University , Xi'an , People's Republic of China
| | - Xinyu Song
- a College of Mathematics and Information Science , Xinyang Normal University , Xinyang , People's Republic of China
| | - Libin Rong
- c Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
| |
Collapse
|
5
|
Dessalles R, Fromion V, Robert P. A stochastic analysis of autoregulation of gene expression. J Math Biol 2017; 75:1253-83. [PMID: 28289838 DOI: 10.1007/s00285-017-1116-7] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/08/2015] [Revised: 12/14/2016] [Indexed: 01/15/2023]
Abstract
This paper analyzes, in the context of a prokaryotic cell, the stochastic variability of the number of proteins when there is a control of gene expression by an autoregulation scheme. The goal of this work is to estimate the efficiency of the regulation to limit the fluctuations of the number of copies of a given protein. The autoregulation considered in this paper relies mainly on a negative feedback: the proteins are repressors of their own gene expression. The efficiency of a production process without feedback control is compared to a production process with an autoregulation of the gene expression assuming that both of them produce the same average number of proteins. The main characteristic used for the comparison is the standard deviation of the number of proteins at equilibrium. With a Markovian representation and a simple model of repression, we prove that, under a scaling regime, the repression mechanism follows a Hill repression scheme with an hyperbolic control. An explicit asymptotic expression of the variance of the number of proteins under this regulation mechanism is obtained. Simulations are used to study other aspects of autoregulation such as the rate of convergence to equilibrium of the production process and the case where the control of the production process of proteins is achieved via the inhibition of mRNAs.
Collapse
|
6
|
Alshorman A, Samarasinghe C, Lu W, Rong L. An HIV model with age-structured latently infected cells. J Biol Dyn 2017; 11:192-215. [PMID: 27338168 DOI: 10.1080/17513758.2016.1198835] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
HIV latency remains a major obstacle to viral elimination. The activation rate of latently infected cells may depend on the age of latent infection. In this paper, we develop a model of HIV infection including age-structured latently infected cells. We mathematically analyse the model and use numerical simulations with different activation functions to show that the model can explain the persistence of low-level viremia and the latent reservoir stability in patients on therapy. Sensitivity tests suggest that the model is robust to the changes of most parameters but is sensitive to the relative magnitude of the net generation rate and the long-term activation rate of latently infected cells. To reduce the sensitivity, we extend the model to include homeostatic proliferation of latently infected cells. The new model is robust in reproducing the long-term dynamics of the virus and latently infected cells observed in patients receiving prolonged combination therapy.
Collapse
Affiliation(s)
- Areej Alshorman
- a Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
| | - Chathuri Samarasinghe
- a Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
| | - Wenlian Lu
- b School of Mathematical Science , Fudan University , Shanghai , People's Republic of China
| | - Libin Rong
- a Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
| |
Collapse
|
7
|
Abstract
Time delays can affect the dynamics of HIV infection predicted by mathematical models. In this paper, we studied two mathematical models each with two time delays. In the first model with HIV latency, one delay is the time between viral entry into a cell and the establishment of HIV latency, and the other delay is the time between cell infection and viral production. We defined the basic reproductive number and showed the local and global stability of the steady states. Numerical simulations were performed to evaluate the influence of time delays on the dynamics. In the second model with HIV immune response, one delay is the time between cell infection and viral production, and the other delay is the time needed for the adaptive immune response to emerge to control viral replication. With two positive delays, we obtained the stability crossing curves for the model, which were shown to be a series of open-ended curves.
Collapse
Affiliation(s)
- Areej Alshorman
- a Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
| | - Xia Wang
- b College of Mathematics and Information Science , Xinyang Normal University , Xinyang , People's Republic of China
| | - M Joseph Meyer
- a Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
| | - Libin Rong
- a Department of Mathematics and Statistics , Oakland University , Rochester , MI , USA
| |
Collapse
|