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Mathematics
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Stochastic Models in Mathematical Biology
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Stochastic Models in Mathematical Biology

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Biology".

Deadline for manuscript submissions: closed (20 January 2023) | Viewed by 9370

Special Issue Editor

Dr. Ivo Siekmann

E-Mail Website
Guest Editor
School of Computer Science and Mathematics, Liverpool John Moores University, Liverpool L3 3AF, UK
Interests: mathematical physiology; mathematical ecology; mathematical epidemiology; stochastic modelling; Markov Chain Monte Carlo (MCMC)
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Is biology deterministic or stochastic? Independent from the answer to this question, stochastic models have been applied with great success in all areas of mathematical biology. Applications extend beyond examples such as evolutionary processes or dynamics of biomolecules where the underlying dynamics themselves are considered stochastic.

For example, in population dynamics, stochastic models are used for describing population-level effects of processes occurring at the level of individuals. Examples include the description of stochastic extinction via branching processes and models for dispersal based on diffusion models.

Many biological systems, including ecosystems or neural networks in the brain, are influenced by a variety of processes—each of which, by itself, only has a small impact. The collective impact of these perturbations can be modelled by stochastic differential equations (SDEs), which enable us to study the dynamics of a deterministic process under stochastic fluctuations.

Finally, stochastic models enable us to represent parameter uncertainties in data-driven models obtained using a Bayesian statistics approach or to account for incomplete knowledge, for example, when considering a heterogeneous population of patients.

For this Special Issue, we are particularly interested in new approaches for applying stochastic models in biology as well as original applications of existing frameworks. We also invite manuscripts from the emerging field of parameterising stochastic models with experimental data.

Dr. Ivo Siekmann
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • population dynamics
  • mathematical physiology
  • biophysics
  • signalling
  • Markov models
  • Markov decision processes
  • stochastic differential equations
  • branching processes
  • parametrisation of stochastic models
  • Bayesian analysis

Related Special Issue

Published Papers (6 papers)

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Research

17 pages, 2005 KiB  
Article
A Stochastic Model of Personality Differences Based on PSI Theory
by Molly Hoy, Sarah Fritsch, Thomas Bröcker, Julius Kuhl and Ivo Siekmann
Mathematics 2023, 11(5), 1182; https://0-doi-org.brum.beds.ac.uk/10.3390/math11051182 - 28 Feb 2023
Viewed by 1564
Abstract
Personality Systems Interactions (PSI) theory explains differences in personality based on the properties of four cognitive systems—object recognition (OR), intuitive behaviour (IB), intention memory (IM) and extension memory (EM). Each system is associated with characteristic modes of perception and behaviour, so personality is [...] Read more.
Personality Systems Interactions (PSI) theory explains differences in personality based on the properties of four cognitive systems—object recognition (OR), intuitive behaviour (IB), intention memory (IM) and extension memory (EM). Each system is associated with characteristic modes of perception and behaviour, so personality is determined by which systems are primarily active. According to PSI theory, the activities of the cognitive systems are regulated by positive and negative affect (reward and punishment). Thus, differences in personality ultimately emerge from four parameters—the sensitivities of up- or downregulating positive and negative affect. The complex interactions of affect and cognitive systems have been represented in a mathematical model based on a system of differential equations. In this study, the environment of a person represented by the mathematical model is modelled by a time series of perturbations with positive and negative affect that are generated by a stochastic process. Comparing the average activities of the cognitive systems for different parameter sets exposed to the same time series of affect perturbations, we observe that different dominant cognitive systems emerge. This demonstrates that different sensitivities for positive and negative affect lead to different modes of cognition and, thus, to different personality types such as agreeable, conscientious, self-determined or independent. Varying the relative frequencies of negative and positive affect perturbations reveals that the average activities of all cognitive systems respond linearly. This observation enables us to predict that conscientious and independent personalities benefit from increased exposure to positive affect, whereas agreeable and self-determined personalities achieve a better balance of their cognitive systems by increased negative affect. Full article
(This article belongs to the Special Issue Stochastic Models in Mathematical Biology)
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14 pages, 9874 KiB  
Article
Stochastic Modeling of Within-Host Dynamics of Plasmodium Falciparum
by Xiao Sun, James M. McCaw and Pengxing Cao
Mathematics 2022, 10(21), 4057; https://0-doi-org.brum.beds.ac.uk/10.3390/math10214057 - 01 Nov 2022
Cited by 1 | Viewed by 1136
Abstract
Malaria remains a major public health burden in South-East Asia and Africa. Mathematical models of within-host infection dynamics and drug action, developed in support of malaria elimination initiatives, have significantly advanced our understanding of the dynamics of infection and supported development of effective [...] Read more.
Malaria remains a major public health burden in South-East Asia and Africa. Mathematical models of within-host infection dynamics and drug action, developed in support of malaria elimination initiatives, have significantly advanced our understanding of the dynamics of infection and supported development of effective drug-treatment regimens. However, the mathematical models supporting these initiatives are predominately based on deterministic dynamics and therefore cannot capture stochastic phenomena such as extinction (no parasitized red blood cells) following treatment, with potential consequences for our interpretation of data sets in which recrudescence is observed. Here we develop a stochastic within-host infection model to study the growth, decline and possible stochastic extinction of parasitized red blood cells in malaria-infected human volunteers. We show that stochastic extinction can occur when the inoculation size is small or when the number of parasitized red blood cells reduces significantly after an antimalarial treatment. We further show that the drug related parameters, such as the maximum killing rate and half-maximum effective concentration, are the primary factors determining the probability of stochastic extinction following treatment, highlighting the importance of highly-efficacious antimalarials in increasing the probability of cure for the treatment of malaria patients. Full article
(This article belongs to the Special Issue Stochastic Models in Mathematical Biology)
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12 pages, 861 KiB  
Article
Analysis of a Class of Predation-Predation Model Dynamics with Random Perturbations
by Xuewen Tan, Pengpeng Liu, Wenhui Luo and Hui Chen
Mathematics 2022, 10(18), 3238; https://0-doi-org.brum.beds.ac.uk/10.3390/math10183238 - 06 Sep 2022
Viewed by 936
Abstract
In this paper, we study a class of predation–prey biological models with random perturbation. Firstly, the existence and uniqueness of systematic solutions can be proven according to Lipschitz conditions, and then we prove that the systematic solution exists globally. Moreover, the article discusses [...] Read more.
In this paper, we study a class of predation–prey biological models with random perturbation. Firstly, the existence and uniqueness of systematic solutions can be proven according to Lipschitz conditions, and then we prove that the systematic solution exists globally. Moreover, the article discusses the long-term dynamical behavior of the model, which studies the stationary distribution and gradual properties of the system. Next, we use two different methods to give the conditions of population extinction. From what has been discussed above, we can safely draw the conclusion that our results are reasonable by using numerical simulation. Full article
(This article belongs to the Special Issue Stochastic Models in Mathematical Biology)
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17 pages, 729 KiB  
Article
Stochastic Epidemic Model for COVID-19 Transmission under Intervention Strategies in China
by Zin Thu Win, Mahmoud A. Eissa and Boping Tian
Mathematics 2022, 10(17), 3119; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173119 - 31 Aug 2022
Cited by 6 | Viewed by 1399
Abstract
In this paper, we discuss an EIQJR model with stochastic perturbation. First, a globally positive solution of the proposed model has been discussed. In addition, the global asymptotic stability and exponential mean-square stability of the disease-free equilibrium have been proven under suitable conditions [...] Read more.
In this paper, we discuss an EIQJR model with stochastic perturbation. First, a globally positive solution of the proposed model has been discussed. In addition, the global asymptotic stability and exponential mean-square stability of the disease-free equilibrium have been proven under suitable conditions for our model. This means that the disease will die over time. We investigate the asymptotic behavior around the endemic equilibrium of the deterministic model to show when the disease will prevail. Constructing a suitable Lyapunov functional method is crucial to our investigation. Parameter estimations and numerical simulations are performed to depict the transmission process of COVID-19 pandemic in China and to support analytical results. Full article
(This article belongs to the Special Issue Stochastic Models in Mathematical Biology)
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11 pages, 887 KiB  
Article
Stationary Probability Density Analysis for the Randomly Forced Phytoplankton–Zooplankton Model with Correlated Colored Noises
by Yuanlin Ma and Xingwang Yu
Mathematics 2022, 10(14), 2383; https://0-doi-org.brum.beds.ac.uk/10.3390/math10142383 - 07 Jul 2022
Cited by 2 | Viewed by 1285
Abstract
In this paper, we propose a stochastic phytoplankton–zooplankton model driven by correlated colored noises, which contains both anthropogenic and natural toxins. Using Khasminskii transformation and the stochastic averaging method, we first transform the original system into an Itô diffusion system. Afterwards, we derive [...] Read more.
In this paper, we propose a stochastic phytoplankton–zooplankton model driven by correlated colored noises, which contains both anthropogenic and natural toxins. Using Khasminskii transformation and the stochastic averaging method, we first transform the original system into an Itô diffusion system. Afterwards, we derive the stationary probability density of the averaging amplitude equation by utilizing the corresponding Fokker–Planck–Kolmogorov equation. Then, the stability of the averaging amplitude is studied and the joint probability density of the original two-dimensional system is given. Finally, the theoretical results are verified by numerical simulations, and the effects of noise characteristics and toxins on system dynamics are further illustrated. Full article
(This article belongs to the Special Issue Stochastic Models in Mathematical Biology)
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8 pages, 339 KiB  
Article
Asymptotic Properties of One Mathematical Model in Food Engineering under Stochastic Perturbations
by Leonid Shaikhet
Mathematics 2021, 9(23), 3013; https://0-doi-org.brum.beds.ac.uk/10.3390/math9233013 - 24 Nov 2021
Cited by 1 | Viewed by 1044
Abstract
For the example of one nonlinear mathematical model in food engineering with several equilibria and stochastic perturbations, a simple criterion for determining a stable or unstable equilibrium is reported. The obtained analytical results are illustrated by detailed numerical simulations of solutions of the [...] Read more.
For the example of one nonlinear mathematical model in food engineering with several equilibria and stochastic perturbations, a simple criterion for determining a stable or unstable equilibrium is reported. The obtained analytical results are illustrated by detailed numerical simulations of solutions of the considered Ito stochastic differential equations. The proposed criterion can be used for a wide class of nonlinear mathematical models in different applications. Full article
(This article belongs to the Special Issue Stochastic Models in Mathematical Biology)
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