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Mathematics
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Applied Mathematics in Disease Control and Dynamics
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Applied Mathematics in Disease Control and Dynamics

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

Deadline for manuscript submissions: 31 December 2024 | Viewed by 4551

Special Issue Editors

Dr. Vasiliy Leonenko

E-Mail Website
Guest Editor
Faculty of Digital Transformation, ITMO University, Kronverksky Pr. 49A, St. Petersburg 197101, Russia
Interests: mathematical modeling; epidemiology; data science; complex systems; high-performance computing
Dr. Olga Krivorotko

E-Mail Website
Guest Editor
Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of Sciences
Interests: inverse and ill-posed problems; epidemiology; immunology; population biology; mean field games; identifiability; optimization; regularization; high-performance computing

Special Issue Information

Dear Colleagues,

Applied mathematics is an important tool in epidemiology because it allows data to be analyzed and predicted for the spread of infectious diseases. With the help of mathematical models, it is possible to estimate the rate of infection spread, determine the most vulnerable groups of the population and develop effective measures for the prevention and control of infectious diseases. Thus, applied mathematics plays an important role in epidemiology and makes it possible to fight infectious diseases more effectively, raising the population’s standards of living and improving the quality of medical services.

This Special Issue focuses on current advances in mathematical epidemiology. It provides a platform for researchers from both academia and industry to present their novel and unpublished work in this domain, regarding (but not limited to) the topics of mathematical modeling and disease forecasting, evaluating epidemiological parameters, planning the measures of disease control and assessing their effectiveness.

Dr. Vasiliy Leonenko
Dr. Olga Krivorotko
Guest Editors

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

  • epidemiology
  • mathematical modeling
  • statistics
  • forecasting
  • disease control

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Published Papers (7 papers)

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Research

28 pages, 508 KiB  
Article
Exploring a Mathematical Model with Saturated Treatment for the Co-Dynamics of Tuberculosis and Diabetes
by Saburi Rasheed, Olaniyi S. Iyiola, Segun I. Oke and Bruce A. Wade
Mathematics 2024, 12(23), 3765; https://doi.org/10.3390/math12233765 - 29 Nov 2024
Viewed by 167
Abstract
In this research, we present a deterministic epidemiological mathematical model that delves into the intricate dynamics of the coexistence of tuberculosis and diabetes. Our comprehensive analysis explores the interplay and the influence of diabetes on tuberculosis incidence within a human population segregated into [...] Read more.
In this research, we present a deterministic epidemiological mathematical model that delves into the intricate dynamics of the coexistence of tuberculosis and diabetes. Our comprehensive analysis explores the interplay and the influence of diabetes on tuberculosis incidence within a human population segregated into diabetic and non-diabetic groups. The model incorporates a saturated incidence rate and treatment regimen for latent tuberculosis infections, offering insights into their impact on tuberculosis control. The theoretical findings reveal the emergence of a phenomenon known as backward bifurcation, attributed to exogenous reinfection and saturated treatment. Additionally, our study employs both local and global sensitivity analyses to identify pivotal parameters crucial to the spread of tuberculosis within the population. This investigation contributes valuable insights to the understanding of the complex relationship between tuberculosis and diabetes, offering a foundation for more effective disease control strategies. Full article
(This article belongs to the Special Issue Applied Mathematics in Disease Control and Dynamics)
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19 pages, 1009 KiB  
Article
Inverse Coefficient Problem for Epidemiological Mean-Field Formulation
by Viktoriya Petrakova
Mathematics 2024, 12(22), 3581; https://doi.org/10.3390/math12223581 - 15 Nov 2024
Viewed by 412
Abstract
The paper proposes an approach to solving the inverse epidemiological problem, written in terms of the “mean-field” theory. Finding the coefficients of an epidemiological SIR mean-field model is reduced to solving an optimization problem, for the solution of which only zero-order methods can [...] Read more.
The paper proposes an approach to solving the inverse epidemiological problem, written in terms of the “mean-field” theory. Finding the coefficients of an epidemiological SIR mean-field model is reduced to solving an optimization problem, for the solution of which only zero-order methods can be used. An algorithm for the solution of the inverse coefficient problem is proposed. Computational experiments were carried out to compare the obtained solutions with respect to synthetic and real data. The results of computational experiments have shown the efficiency of this approach. Ways to further improve the approach have also been determined. Full article
(This article belongs to the Special Issue Applied Mathematics in Disease Control and Dynamics)
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19 pages, 3727 KiB  
Article
Dynamic Programming-Based Approach to Model Antigen-Driven Immune Repertoire Synthesis
by Alexander S. Bratus, Gennady Bocharov and Dmitry Grebennikov
Mathematics 2024, 12(20), 3291; https://doi.org/10.3390/math12203291 - 20 Oct 2024
Viewed by 729
Abstract
This paper presents a novel approach to modeling the repertoire of the immune system and its adaptation in response to the evolutionary dynamics of pathogens associated with their genetic variability. It is based on application of a dynamic programming-based framework to model the [...] Read more.
This paper presents a novel approach to modeling the repertoire of the immune system and its adaptation in response to the evolutionary dynamics of pathogens associated with their genetic variability. It is based on application of a dynamic programming-based framework to model the antigen-driven immune repertoire synthesis. The processes of formation of new receptor specificity of lymphocytes (the growth of their affinity during maturation) are described by an ordinary differential equation (ODE) with a piecewise-constant right-hand side. Optimal control synthesis is based on the solution of the Hamilton–Jacobi–Bellman equation implementing the dynamic programming approach for controlling Gaussian random processes generated by a stochastic differential equation (SDE) with the noise in the form of the Wiener process. The proposed description of the clonal repertoire of the immune system allows us to introduce an integral characteristic of the immune repertoire completeness or the integrative fitness of the whole immune system. The quantitative index for characterizing the immune system fitness is analytically derived using the Feynman–Kac–Kolmogorov equation. Full article
(This article belongs to the Special Issue Applied Mathematics in Disease Control and Dynamics)
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22 pages, 1992 KiB  
Article
The Forecasting of the Spread of Infectious Diseases Based on Conditional Generative Adversarial Networks
by Olga Krivorotko and Nikolay Zyatkov
Mathematics 2024, 12(19), 3044; https://doi.org/10.3390/math12193044 - 28 Sep 2024
Viewed by 553
Abstract
New epidemics encourage the development of new mathematical models of the spread and forecasting of infectious diseases. Statistical epidemiology data are characterized by incomplete and inexact time series, which leads to an unstable and non-unique forecasting of infectious diseases. In this paper, a [...] Read more.
New epidemics encourage the development of new mathematical models of the spread and forecasting of infectious diseases. Statistical epidemiology data are characterized by incomplete and inexact time series, which leads to an unstable and non-unique forecasting of infectious diseases. In this paper, a model of a conditional generative adversarial neural network (CGAN) for modeling and forecasting COVID-19 in St. Petersburg is constructed. It takes 20 processed historical statistics as a condition and is based on the solution of the minimax problem. The CGAN builds a short-term forecast of the number of newly diagnosed COVID-19 cases in the region for 5 days ahead. The CGAN approach allows modeling the distribution of statistical data, which allows obtaining the required amount of training data from the resulting distribution. When comparing the forecasting results with the classical differential SEIR-HCD model and a recurrent neural network with the same input parameters, it was shown that the forecast errors of all three models are in the same range. It is shown that the prediction error of the bagging model based on three models is lower than the results of each model separately. Full article
(This article belongs to the Special Issue Applied Mathematics in Disease Control and Dynamics)
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17 pages, 810 KiB  
Article
Analysis and Optimal Control of a Two-Strain SEIR Epidemic Model with Saturated Treatment Rate
by Yudie Hu, Hongyan Wang and Shaoping Jiang
Mathematics 2024, 12(19), 3026; https://doi.org/10.3390/math12193026 - 27 Sep 2024
Viewed by 635
Abstract
In this paper, we conducted a study on the optimal control problem of an epidemic model which consists of two strain with different types of incidence rates: bilinear and non-monotonic. We also considered use of the saturation treatment function. Two basic regeneration numbers [...] Read more.
In this paper, we conducted a study on the optimal control problem of an epidemic model which consists of two strain with different types of incidence rates: bilinear and non-monotonic. We also considered use of the saturation treatment function. Two basic regeneration numbers are calculated from the epidemic model, which are denoted as R1 and R2. The global stability of the disease-free equilibrium point was studied by the Lyapunov method, and it was proved that the disease-free equilibrium point is globally asymptotically stable when R1 and R2 are less than one. Finally, we formulated a time-dependent optimal control problem by Pontryagin’s maximum principle. Numerical simulations were performed to establish the effects of model parameters for disease transmission as well as the effects of control. Full article
(This article belongs to the Special Issue Applied Mathematics in Disease Control and Dynamics)
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20 pages, 947 KiB  
Article
A Spatial Agent-Based Model for Studying the Effect of Human Mobility Patterns on Epidemic Outbreaks in Urban Areas
by Alexandru Topîrceanu
Mathematics 2024, 12(17), 2765; https://doi.org/10.3390/math12172765 - 6 Sep 2024
Viewed by 605
Abstract
The epidemic outbreaks of the last two decades have led governments to rely more on computational tools for establishing protection policies. Computational approaches to modeling epidemics traditionally rely on compartmental models, network models, or agent-based models (ABMs); however, each approach has its limitations, [...] Read more.
The epidemic outbreaks of the last two decades have led governments to rely more on computational tools for establishing protection policies. Computational approaches to modeling epidemics traditionally rely on compartmental models, network models, or agent-based models (ABMs); however, each approach has its limitations, ranging from reduced realism to lack of tractability. Furthermore, the recent literature emphasizes the importance of points of interest (POIs) as sources of population mixing and potential outbreak hotspots. In response, this study proposes a novel urban spatial ABM validated using our augmented SICARQD epidemic model. To replicate daily activities more accurately, the urban area is divided into a matrix of points of interest (POIs) with agents that have unique paths that only permit infectious transmission within POIs. Our results provide a qualitative assessment of how urban characteristics and individual mobility patterns impact the infected population during an outbreak. That is, we study how population density, the total number of POIs (where the population concentrates), the average number of POIs visited by an agent, the maximum travel distance from the home location, and the quarantine ratio impact the dynamics of an outbreak. Our ABM simulation framework offers a valuable tool for investigating and controlling infectious disease outbreaks in urban environments with direct applicability to global policy makers. Full article
(This article belongs to the Special Issue Applied Mathematics in Disease Control and Dynamics)
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21 pages, 3442 KiB  
Article
Parameter Tuning of Agent-Based Models: Metaheuristic Algorithms
by Andrei I. Vlad, Alexei A. Romanyukha and Tatiana E. Sannikova
Mathematics 2024, 12(14), 2208; https://doi.org/10.3390/math12142208 - 15 Jul 2024
Viewed by 733
Abstract
When it comes to modelling complex systems using an agent-based approach, there is a problem of choosing the appropriate parameter optimisation technique. This problem is further aggravated by the fact that the parameter space in complex agent-based systems can have a large dimension, [...] Read more.
When it comes to modelling complex systems using an agent-based approach, there is a problem of choosing the appropriate parameter optimisation technique. This problem is further aggravated by the fact that the parameter space in complex agent-based systems can have a large dimension, and the time required to perform numerical experiments can be large. An alternative approach to traditional optimisation methods are the so-called metaheuristic algorithms, which provide an approximate solution in an acceptable time. The purpose of this study is to compare various metaheuristic algorithms for parameter tuning and to analyse their effectiveness applied to two agent-based models with different complexities. In this study, we considered commonly used metaheuristic algorithms for agent-based model optimisation: the Markov chain Monte Carlo method, the surrogate modelling approach, the particle swarm optimisation algorithm, and the genetic algorithm, as well as the more novel chaos game optimisation algorithm. The proposed algorithms were tested on two agent-based models, one of which was a simple toy model of the spread of contagious disease, and the other was a more complex model of the circulation of respiratory viruses in a city with 10 million agents and 26 calibrated parameters. Full article
(This article belongs to the Special Issue Applied Mathematics in Disease Control and Dynamics)
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