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

Deadline for manuscript submissions: closed (30 November 2021) | Viewed by 34005

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Dr. Mikhail Kolev

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1. Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Słoneczna 54 Street, 10-710 Olsztyn, Poland
2. Department of Mathematics, University of Architecture, Civil Engineering and Geodesy, 1 Hr. Smirnenski Blvd., 1046 Sofia, Bulgaria
Interests: mathematical modeling; numerical methods; statistical analysis; linear regression; programming; biology
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Dear Colleagues,

Over the recent decades, mathematical models have been actively used in various fields of technology and science, both in the natural and in the social sciences.

An important domain of their application is the mathematical modeling of complex systems, and in particular living systems. Real experiments in some cases cannot be conducted on living beings due to the complexity of their organisms or due to the lack of necessary technology in some situations. Such experiments are often long-lasting, expensive, and problematic from an ethical viewpoint.

Mathematical models can describe some characteristic properties of the phenomena under consideration and predict the possible scenarios for their course without the need to conduct real experiments.

Unlike the modeling of physicomechanical systems, scientists dealing with biological systems need to consider the specific differences between living and inert matter. As is well known, systems pertaining to inert matter can be described using invariance principles and conservation laws, and the interactions between their individual elements follow the laws of classical or quantum mechanics. In contrast, in living organisms, these laws cannot be directly applied. Because of their nature and the need to survive, living beings are characterized by high internal complexity. They eat, breathe, protect themselves from pests and predators, and as a result, complex processes of transformation of substances and energy take place.

In the process of centuries of evolution, in their struggle for survival in a variety of conditions, organisms have improved themselves, developing the ability to change the ways of functioning of their constituting elements, and eventually their reproduction or destruction depending on the respective conditions.

Mathematical models have been successfully applied to study various diseases, such as cancer, infectious, autoimmune, cardiovascular, neurodegenerative, and others. The topic is especially relevant in view of the development of COVID‑19.

Mathematical modeling can contribute to the improvement of understanding the role of key factors in various biological processes and phenomena—in particular, the occurrence and development of various diseases in medicine, the improvement of existing and creation of new drugs, the optimization of treatment protocols, and the improvement of hospital technology and effective healthcare system management. The proposed applications of models in biology and medicine can impact the development of mathematical theory and computational methods.

The purpose of this Special Issue is to publish qualitative papers, referring but not limited to the derivation of new and improvement of existing mathematical models, designed at various observation and representation scales, applicable in biology and medicine, their qualitative and quantitative analysis, as well as comparison of the results of the modeling with experimental and clinical data.

Please note that all of the submitted papers must be within the general scope of the Mathematics journal.

Dr. Mikhail Kolev
Guest Editor

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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.

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Keywords

deterministic models
stochastic models
discrete models
continuous models
spatially distributed models
multiscale models
population models
epidemic models
kinetic models
active particles
models with delay

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Published Papers (12 papers)

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Research

25 pages, 1351 KiB

Open AccessFeature PaperArticle

Stability Analysis of an Eight Parameter SIR-Type Model Including Loss of Immunity, and Disease and Vaccination Fatalities

by Florin Avram, Rim Adenane, Gianluca Bianchin and Andrei Halanay

Mathematics 2022, 10(3), 402; https://doi.org/10.3390/math10030402 - 27 Jan 2022

Cited by 8 | Viewed by 2608

Abstract

We revisit here a landmark five-parameter SIR-type model, which is maybe the simplest example where a complete picture of all cases, including non-trivial bistability behavior, may be obtained using simple tools. We also generalize it by adding essential vaccination and vaccination-induced death parameters, with the aim of revealing the role of vaccination and its possible failure. The main result is Theorem 1, which describes the stability behavior of our model in all possible cases. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis in Biology and Medicine)

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22 pages, 430 KiB

Open AccessArticle

An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion

by Mikhail K. Kolev, Miglena N. Koleva and Lubin G. Vulkov

Mathematics 2022, 10(1), 131; https://doi.org/10.3390/math10010131 - 2 Jan 2022

Cited by 6 | Viewed by 1893

Abstract

In this paper, we consider models of cancer migration and invasion, which consist of two nonlinear parabolic equations (one of the convection–diffusion reaction type and the other of the diffusion–reaction type) and an additional nonlinear ordinary differential equation. The unknowns represent concentrations or densities that cannot be negative. Widely used approximations, such as difference schemes, can produce negative solutions because of truncation errors and can become unstable. We propose a new difference scheme that guarantees the positivity of the numerical solution for arbitrary mesh step sizes. It has explicit and fast performance even for nonlinear reaction terms that consist of sums of positive and negative functions. The numerical examples illustrate the simplicity and efficiency of the method. A numerical simulation of a model of cancer migration is also discussed. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis in Biology and Medicine)

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22 pages, 727 KiB

Open AccessArticle

The Generalized DUS Transformed Log-Normal Distribution and Its Applications to Cancer and Heart Transplant Datasets

by Muhammed Rasheed Irshad, Christophe Chesneau, Soman Latha Nitin, Damodaran Santhamani Shibu and Radhakumari Maya

Mathematics 2021, 9(23), 3113; https://doi.org/10.3390/math9233113 - 2 Dec 2021

Cited by 3 | Viewed by 1739

Abstract

Many studies have underlined the importance of the log-normal distribution in the modeling of phenomena occurring in biology. With this in mind, in this article we offer a new and motivated transformed version of the log-normal distribution, primarily for use with biological data. The hazard rate function, quantile function, and several other significant aspects of the new distribution are investigated. In particular, we show that the hazard rate function has increasing, decreasing, bathtub, and upside-down bathtub shapes. The maximum likelihood and Bayesian techniques are both used to estimate unknown parameters. Based on the proposed distribution, we also present a parametric regression model and a Bayesian regression approach. As an assessment of the longstanding performance, simulation studies based on maximum likelihood and Bayesian techniques of estimation procedures are also conducted. Two real datasets are used to demonstrate the applicability of the new distribution. The efficiency of the third parameter in the new model is tested by utilizing the likelihood ratio test. Furthermore, the parametric bootstrap approach is used to determine the effectiveness of the suggested model for the datasets. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis in Biology and Medicine)

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15 pages, 663 KiB

Open AccessArticle

A Mathematical Model to Control the Prevalence of a Directly and Indirectly Transmitted Disease

by Begoña Cantó, Carmen Coll, Maria Jesús Pagán, Joan Poveda and Elena Sánchez

Mathematics 2021, 9(20), 2562; https://doi.org/10.3390/math9202562 - 13 Oct 2021

Viewed by 1783

Abstract

In this paper, a mathematical model to describe the spread of an infectious disease on a farm is developed. To analyze the evolution of the infection, the direct transmission from infected individuals and the indirect transmission from the bacteria accumulated in the enclosure are considered. A threshold value of population is obtained to assure the extinction of the disease. When this size of population is exceeded, two control procedures to apply at each time are proposed. For each of them, a maximum number of steps without control and reducing the prevalence of disease is obtained. In addition, a criterion to choose between both procedures is established. Finally, the results are numerically simulated for a hypothetical outbreak on a farm. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis in Biology and Medicine)

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10 pages, 409 KiB

Open AccessArticle

Continuum Modelling for Encapsulation of Anticancer Drugs inside Nanotubes

by Mansoor H. Alshehri

Mathematics 2021, 9(19), 2469; https://doi.org/10.3390/math9192469 - 3 Oct 2021

Cited by 7 | Viewed by 1754

Abstract

Nanotubes, such as those made of carbon, silicon, and boron nitride, have attracted tremendous interest in the research community and represent the starting point for the development of nanotechnology. In the current study, the use of nanotubes as a means of drug delivery and, more specifically, for cancer therapy, is investigated. Using traditional applied mathematical modelling, I derive explicit analytical expressions to understand the encapsulation behaviour of drug molecules into different types of single-walled nanotubes. The interaction energies between three anticancer drugs, namely, cisplatin, carboplatin, and doxorubicin, and the nanotubes are observed by adopting the Lennard–Jones potential function together with the continuum approach. This study is focused on determining a favourable size and an appropriate type of nanotube to encapsulate anticancer drugs. The results indicate that the drug molecules with a large size tend to be located inside a large nanotube and that encapsulation depends on the radius and type of the tube. For the three nanotubes used to encapsulate drugs, the results show that the nanotube radius must be at least 5.493 Å for cisplatin, 6.452 Å for carboplatin, and 10.208 Å for doxorubicin, and the appropriate type to encapsulate drugs is the boron nitride nanotube. There are some advantages to using different types of nanotubes as a means of drug delivery, such as improved chemical stability, reduced synthesis costs, and improved biocompatibility. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis in Biology and Medicine)

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19 pages, 2617 KiB

Open AccessArticle

A Dynamic Model of Cytosolic Calcium Concentration Oscillations in Mast Cells

by Mingzhu Sun, Yingchen Li and Wei Yao

Mathematics 2021, 9(18), 2322; https://doi.org/10.3390/math9182322 - 19 Sep 2021

Cited by 4 | Viewed by 2554

Abstract

In this paper, a dynamic model of cytosolic calcium concentration (

{{[Ca}^{2 +}]}_{Cyt}

) oscillations is established for mast cells (MCs). This model includes the cytoplasm (Cyt), endoplasmic reticulum (ER), mitochondria (Mt), and functional region (μd), formed by the [...] Read more.

In this paper, a dynamic model of cytosolic calcium concentration (

{{[Ca}^{2 +}]}_{Cyt}

) oscillations is established for mast cells (MCs). This model includes the cytoplasm (Cyt), endoplasmic reticulum (ER), mitochondria (Mt), and functional region (μd), formed by the ER and Mt, also with

{Ca}^{2 +}

channels in these cellular compartments. By this model, we calculate

{{[Ca}^{2 +}]}_{Cyt}

oscillations that are driven by distinct mechanisms at varying

k_{\deg}

(degradation coefficient of inositol 1,4,5-trisphosphate,

{IP}_{3}

and production coefficient of

{IP}_{3}

), as well as at different distances between the ER and Mt (ER–Mt distance). The model predicts that (i) Mt and μd compartments can reduce the amplitude of

{{[Ca}^{2 +}]}_{Cyt}

oscillations, and cause the ER to release less

{Ca}^{2 +}

during oscillations; (ii) with increasing cytosolic

{IP}_{3}

concentration (

{{[IP}_{3}]}_{Cyt}

), the amplitude of oscillations increases (from 0.1 μM to several μM), but the frequency decreases; (iii) the frequency of

{{[Ca}^{2 +}]}_{Cyt}

oscillations decreases as the ER–Mt distance increases. What is more, when the ER–Mt distance is greater than 65 nm, the μd compartment has less effect on

{{[Ca}^{2 +}]}_{Cyt}

oscillations. These results suggest that Mt, μd, and

{IP}_{3}

can all affect the amplitude and frequency of

{{[Ca}^{2 +}]}_{Cyt}

oscillations, but the mechanism is different. The model provides a comprehensive mechanism for predicting cytosolic

{Ca}^{2 +}

concentration oscillations in mast cells, and a theoretical basis for calcium oscillations observed in mast cells, so as to better understand the regulation mechanism of calcium signaling in mast cells. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis in Biology and Medicine)

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17 pages, 3937 KiB

Open AccessArticle

Nonlinear Combinational Dynamic Transmission Rate Model and Its Application in Global COVID-19 Epidemic Prediction and Analysis

by Xiaojin Xie, Kangyang Luo, Zhixiang Yin and Guoqiang Wang

Mathematics 2021, 9(18), 2307; https://doi.org/10.3390/math9182307 - 18 Sep 2021

Cited by 4 | Viewed by 1743

Abstract

The outbreak of coronavirus disease 2019 (COVID-19) has caused a global disaster, seriously endangering human health and the stability of social order. The purpose of this study is to construct a nonlinear combinational dynamic transmission rate model with automatic selection based on forecasting effective measure (FEM) and support vector regression (SVR) to overcome the shortcomings of the difficulty in accurately estimating the basic infection number

R_{0}

and the low accuracy of single model predictions. We apply the model to analyze and predict the COVID-19 outbreak in different countries. First, the discrete values of the dynamic transmission rate are calculated. Second, the prediction abilities of all single models are comprehensively considered, and the best sliding window period is derived. Then, based on FEM, the optimal sub-model is selected, and the prediction results are nonlinearly combined. Finally, a nonlinear combinational dynamic transmission rate model is developed to analyze and predict the COVID-19 epidemic in the United States, Canada, Germany, Italy, France, Spain, South Korea, and Iran in the global pandemic. The experimental results show an the out-of-sample forecasting average error rate lower than 10.07% was achieved by our model, the prediction of COVID-19 epidemic inflection points in most countries shows good agreement with the real data. In addition, our model has good anti-noise ability and stability when dealing with data fluctuations. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis in Biology and Medicine)

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18 pages, 651 KiB

Open AccessArticle

Mathematical Study for Chikungunya Virus with Nonlinear General Incidence Rate

by Salah Alsahafi and Stephen Woodcock

Mathematics 2021, 9(18), 2186; https://doi.org/10.3390/math9182186 - 7 Sep 2021

Cited by 5 | Viewed by 1887

Abstract

In this article, we examine the dynamics of a Chikungunya virus (CHIKV) infection model with two routes of infection. The model uses four categories, namely, uninfected cells, infected cells, the CHIKV virus, and antibodies. The equilibrium points of the model, which consist of the free point for the CHIKV and CHIKV endemic point, are first analytically determined. Next, the local stability of the equilibrium points is studied, based on the basic reproduction number (

R_{0}

) obtained by the next-generation matrix. From the analysis, it is found that the disease-free point is locally asymptotically stable if

R_{0} \leq 1

, and the CHIKV endemic point is locally asymptotically stable if

R_{0} > 1

. Using the Lyapunov method, the global stability analysis of the steady-states confirms the local stability results. We then describe our design of an optimal recruitment strategy to minimize the number of infected cells, as well as a nonlinear optimal control problem. Some numerical simulations are provided to visualize the analytical results obtained. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis in Biology and Medicine)

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26 pages, 638 KiB

Open AccessArticle

A Novel Technique to Control the Accuracy of a Nonlinear Fractional Order Model of COVID-19: Application of the CESTAC Method and the CADNA Library

by Samad Noeiaghdam, Sanda Micula and Juan J. Nieto

Mathematics 2021, 9(12), 1321; https://doi.org/10.3390/math9121321 - 8 Jun 2021

Cited by 39 | Viewed by 4292

Abstract

In this paper, a nonlinear fractional order model of COVID-19 is approximated. For this aim, at first we apply the Caputo–Fabrizio fractional derivative to model the usual form of the phenomenon. In order to show the existence of a solution, the Banach fixed point theorem and the Picard–Lindelof approach are used. Additionally, the stability analysis is discussed using the fixed point theorem. The model is approximated based on Indian data and using the homotopy analysis transform method (HATM), which is among the most famous, flexible and applicable semi-analytical methods. After that, the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method and the CADNA (Control of Accuracy and Debugging for Numerical Applications) library, which are based on discrete stochastic arithmetic (DSA), are applied to validate the numerical results of the HATM. Additionally, the stopping condition in the numerical algorithm is based on two successive approximations and the main theorem of the CESTAC method can aid us analytically to apply the new terminations criterion instead of the usual absolute error that we use in the floating-point arithmetic (FPA). Finding the optimal approximations and the optimal iteration of the HATM to solve the nonlinear fractional order model of COVID-19 are the main novelties of this study. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis in Biology and Medicine)

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35 pages, 42344 KiB

Open AccessArticle

A Stochastic Kinetic Type Reactions Model for COVID-19

by Giorgio Sonnino, Fernando Mora and Pasquale Nardone

Mathematics 2021, 9(11), 1221; https://doi.org/10.3390/math9111221 - 27 May 2021

Cited by 2 | Viewed by 3124

Abstract

We propose two stochastic models for the Coronavirus pandemic. The statistical properties of the models, in particular the correlation functions and the probability density functions, were duly computed. Our models take into account the adoption of lockdown measures as well as the crucial role of hospitals and health care institutes. To accomplish this work we adopt a kinetic-type reaction approach where the modelling of the lockdown measures is obtained by introducing a new mathematical basis and the intensity of the stochastic noise is derived by statistical mechanics. We analysed two scenarios: the stochastic

S I S

-model (Susceptible ⇒ Infectious ⇒ Susceptible) and the stochastic

S I S

-model integrated with the action of the hospitals; both models take into account the lockdown measures. We show that, for the case of the stochastic SIS-model, once the lockdown measures are removed, the Coronavirus infection will start growing again. However, the combined contributions of lockdown measures with the action of hospitals and health institutes is able to contain and even to dampen the spread of the SARS-CoV-2 epidemic. This result may be used during a period of time when the massive distribution of vaccines in a given population is not yet feasible. We analysed data for USA and France. In the case of USA, we analysed the following situations: USA is subjected to the first wave of infection by Coronavirus and USA is in the second wave of SARS-CoV-2 infection. The agreement between theoretical predictions and real data confirms the validity of our approach. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis in Biology and Medicine)

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16 pages, 1161 KiB

Open AccessArticle

Age-Specific Mathematical Model for Tuberculosis Transmission Dynamics in South Korea

by Sunmi Lee, Hae-Young Park, Hohyung Ryu and Jin-Won Kwon

Mathematics 2021, 9(8), 804; https://doi.org/10.3390/math9080804 - 7 Apr 2021

Cited by 7 | Viewed by 4180

Abstract

Korea has the highest burden of tuberculosis (TB) among Organization for Economic Co-operation and Development countries. Various strategies have been implemented to eradicate TB in Korea, and it is critical to evaluate previous TB management outcomes before framing future TB policies. Over the past few decades, the rapid increase in the aging population in Korea has substantially impacted the incidence of TB among the elderly. Thus, in this study, we aimed to develop a mathematical model for the assessment of TB management outcomes incorporating special features of TB transmission dynamics in Korea. First, we incorporate 2-age groups in our TB model because TB epidemics in Korea are different between the elderly and the non-elderly (<65 years vs. ≥65 years). Second, because the public-private mix has had a full-fledged impact since 2012, this study was divided into two periods (2001–2011 and 2012–2018). We developed a mathematical model of TB transmission dynamics with 2-age groups and age-specific model parameters were estimated based on actual TB epidemic data from 2001 to 2018. These parameters included transmission rates, relapse rates, and recovery rates. We conducted sensitivity analyses of various parameters, and investigated the impacts of these parameters on TB incidence. Our results demonstrate that the overall outcomes of both age-groups improved in the period of 2012–2018 compared with that in the period of 2001–2011. Age-specific interventions should be implemented to reduce the overall TB incidence. More intensive treatment efforts should be focused on the elderly, while the early detection and treatment rates for latent TB were the most significant factor to reduce TB incidence in both groups. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis in Biology and Medicine)

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16 pages, 2077 KiB

Open AccessArticle

Machine Learning Data Imputation and Prediction of Foraging Group Size in a Kleptoparasitic Spider

by Yong-Chao Su, Cheng-Yu Wu, Cheng-Hong Yang, Bo-Sheng Li, Sin-Hua Moi and Yu-Da Lin

Mathematics 2021, 9(4), 415; https://doi.org/10.3390/math9040415 - 20 Feb 2021

Cited by 6 | Viewed by 2475

Abstract

Cost–benefit analysis is widely used to elucidate the association between foraging group size and resource size. Despite advances in the development of theoretical frameworks, however, the empirical systems used for testing are hindered by the vagaries of field surveys and incomplete data. This study developed the three approaches to data imputation based on machine learning (ML) algorithms with the aim of rescuing valuable field data. Using 163 host spider webs (132 complete data and 31 incomplete data), our results indicated that the data imputation based on random forest algorithm outperformed classification and regression trees, the k-nearest neighbor, and other conventional approaches (Wilcoxon signed-rank test and correlation difference have p-value from < 0.001–0.030). We then used rescued data based on a natural system involving kleptoparasitic spiders from Taiwan and Vietnam (Argyrodes miniaceus, Theridiidae) to test the occurrence and group size of kleptoparasites in natural populations. Our partial least-squares path modelling (PLS-PM) results demonstrated that the size of the host web (T = 6.890, p = 0.000) is a significant feature affecting group size. The resource size (T = 2.590, p = 0.010) and the microclimate (T = 3.230, p = 0.001) are significant features affecting the presence of kleptoparasites. The test of conformation of group size distribution to the ideal free distribution (IFD) model revealed that predictions pertaining to per-capita resource size were underestimated (bootstrap resampling mean slopes <IFD predicted slopes, p < 0.001). These findings highlight the importance of applying appropriate ML methods to the handling of missing field data. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis in Biology and Medicine)

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