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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 (25 April 2023) | Viewed by 19592

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Prof. Dr. Yonghui Xia

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Guest Editor

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Interests: differential equations includes theory of the Quaternion Differential Equations, Hartman-Grobman linearization and spectrum; travelling waves of PDE models; qualitative theory of ODEs such as limit cycles and periodic solution; stability analysis in nonlinear systems, mathematical biology, neural networks (including continuous, discrete and impulsive systems)

Prof. Dr. Youhua Qian

E-Mail Website
Guest Editor

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Interests: homotopy analysis method; chaotic theory and bifurcation theory; bursting oscillation
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In the last few decades, theories of the ordinary or partial differential equations has become a rapidly growing area of research. The attractiveness of this field not only derives from theoretical interests, but also differential equations, which have many applications in several phenomena observed in applied sciences. In particular, mathematical biology has attracted many researchers’ interests. We invite researchers to submit original research articles as well as review articles on theory of ordinary or partial differential equations and their applications to mathematical biology. Potential topics included, but are not limited to:

Special orbits for ordinary differential equations and their applications to mathematical biology;
Lyapunov stability for conservative and dissipative systems and their applications to mathematical biology;
Dichotomy and dynamical spectrum;
Topological linearization and topological conjugacy;
Stability of differential, functional and difference equations and their applications to mathematical biology;
Qualitative theory of solutions of dynamic systems and their applications to mathematical biology;
Periodic and almost periodic solutions of differential, functional, impulsive, and difference equations and their applications to mathematical biology;
Periodic and almost periodic solutions of neutral equations;
Bifurcations and chaos with their applications to mathematical biology.

Prof. Dr. Yonghui Xia
Prof. Dr. Youhua Qian
Guest Editors

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Keywords

stability
bifurcations and chaos
mathematical biology
periodic and almost periodic solutions
dichotomy and dynamical spectrum
topological conjugacy

Published Papers (13 papers)

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Research

15 pages, 318 KiB

Open AccessFeature PaperArticle

Non-Standard and Null Lagrangians for Nonlinear Dynamical Systems and Their Role in Population Dynamics

by Diana T. Pham and Zdzislaw E. Musielak

Mathematics 2023, 11(12), 2671; https://0-doi-org.brum.beds.ac.uk/10.3390/math11122671 - 12 Jun 2023

Cited by 1 | Viewed by 1025

Abstract

Non-standard Lagrangians do not display any discernible energy-like terms, yet they give the same equations of motion as standard Lagrangians, which have easily identifiable energy-like terms. A new method to derive non-standard Lagrangians for second-order nonlinear differential equations with damping is developed and the limitations of this method are explored. It is shown that the limitations do not exist only for those nonlinear dynamical systems that can be converted into linear ones. The obtained results are applied to selected population dynamics models for which non-standard Lagrangians and their corresponding null Lagrangians and gauge functions are derived, and their roles in the population dynamics are discussed. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

13 pages, 340 KiB

Open AccessArticle

Coexistence and Replacement of Two Different Maturation Strategies Adopted by a Stage-Structured Population

by Shuyang Xue

Mathematics 2023, 11(10), 2393; https://0-doi-org.brum.beds.ac.uk/10.3390/math11102393 - 22 May 2023

Viewed by 698

Abstract

Maturation strategies play a key role in the survival and development of populations. In response to changes in the external environment and human interventions, populations adopt appropriate maturation strategies. Different maturation strategies can lead to different birth and mortality rates. In this paper, we develop and analyze a stage-structured population model with two maturation strategies to obtain conditions for the coexistence of two maturation strategies and conditions for competitive exclusion. Our results also show that equality of fitness—represented by basic reproductive numbers being greater than 1 under different maturation strategies—promotes the coexistence of the two strategies. The reason why a strategy is replaced by another one is that the population adopting this strategy has weak fitness, which is measured by the basic reproductive number. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

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20 pages, 2285 KiB

Open AccessArticle

Periodic Behaviour of an Epidemic in a Seasonal Environment with Vaccination

by Miled El Hajji, Dalal M. Alshaikh and Nada A. Almuallem

Mathematics 2023, 11(10), 2350; https://0-doi-org.brum.beds.ac.uk/10.3390/math11102350 - 18 May 2023

Cited by 7 | Viewed by 1224

Abstract

Infectious diseases include all diseases caused by the transmission of a pathogenic agent such as bacteria, viruses, parasites, prions, and fungi. They, therefore, cover a wide spectrum of benign pathologies such as colds or angina but also very serious ones such as AIDS, hepatitis, malaria, or tuberculosis. Many epidemic diseases exhibit seasonal peak periods. Studying the population behaviours due to seasonal environment becomes a necessity for predicting the risk of disease transmission and trying to control it. In this work, we considered a five-dimensional system for a fatal disease in a seasonal environment. We studied, in the first step, the autonomous system by investigating the global stability of the steady states. In a second step, we established the existence, uniqueness, positivity, and boundedness of a periodic orbit. We showed that the global dynamics are determined using the basic reproduction number denoted by

R_{0}

and calculated using the spectral radius of an integral operator. The global stability of the disease-free periodic solution was satisfied if

R_{0} < 1

, and we show also the persistence of the disease once

R_{0} > 1

. Finally, we displayed some numerical investigations supporting the theoretical findings, where the trajectories converge to a limit cycle if

R_{0} > 1

. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

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17 pages, 441 KiB

Open AccessArticle

Mathematical Model to Understand the Dynamics of Cancer, Prevention Diagnosis and Therapy

by Ebraheem Alzahrani, M. M. El-Dessoky and Muhammad Altaf Khan

Mathematics 2023, 11(9), 1975; https://0-doi-org.brum.beds.ac.uk/10.3390/math11091975 - 22 Apr 2023

Cited by 3 | Viewed by 1882

Abstract

In the present study, we formulate a mathematical model to understand breast cancer in the population of Saudi Arabia. We consider a mathematical model and study its mathematical results. We show that the breast cancer model possesses a unique system of solutions. The stability results are shown for the model. We consider the reported cases in Saudi Arabia for the period 2004–2016. The data are given for the female population in Saudi Arabia that is suffering from breast cancer. The data are used to obtain the values of the parameters, and then we predict the long-term behavior with the obtained numerical results. The numerical results are obtained using the proposed parameterized approach. We present graphical results for the breast cancer model under effective parameters such as

τ_{1}

τ_{2}

, and

τ_{3}

that cause decreasing future cases in the population of stages 3 and 4, and the disease-free condition. Chemotherapy generally increases the risk of cardiotoxicity, and, hence, our model result shows this fact. The combination of chemotherapy stages 3 and 4 and the parameters

τ_{1}

and

τ_{2}

together at a low-level rate and also treating the patients before the chemotherapy will decrease the population of cardiotoxicity. The findings of this study are intended to reduce the number of cardiotoxic patients and raise the number of patients who recover following chemotherapy, which will aid in public health decision making. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

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20 pages, 1301 KiB

Open AccessArticle

Stability and Threshold Dynamics in a Seasonal Mathematical Model for Measles Outbreaks with Double-Dose Vaccination

by Mahmoud A. Ibrahim and Attila Dénes

Mathematics 2023, 11(8), 1791; https://0-doi-org.brum.beds.ac.uk/10.3390/math11081791 - 9 Apr 2023

Cited by 5 | Viewed by 1768

Abstract

Measles is a highly contagious viral disease that can lead to serious complications, including death, particularly in young children. In this study, we developed a mathematical model that incorporates a seasonal transmission parameter to examine the measles transmission dynamics. We define the basic [...] Read more.

R_{0}

) and show its utility as a threshold parameter for global dynamics and the existence of periodic solutions. The model was applied to the measles outbreak that occurred in Pakistan from 2019 to 2021 and provided a good fit to the observed data. Our estimate of the basic reproduction number was found to be greater than one, indicating that the disease will persist in the population. The findings highlight the need to increase vaccination coverage and efficacy to mitigate the impact of the epidemic. The model also shows the long-term behavior of the disease, which becomes endemic and recurs annually. Our simulations demonstrate that a shorter incubation period accelerates the spread of the disease, while a higher vaccination coverage rate reduces its impact. The importance of the second dose of the measles vaccine is emphasized, and a higher vaccine efficacy rate can also help bring

R_{0}

below one. Our study provides valuable information for the development and implementation of effective measles control strategies. To prevent future outbreaks, increasing vaccination coverage among the population is the most effective way to reduce the transmission of measles. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

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23 pages, 753 KiB

Open AccessArticle

Viral Infection Spreading in Cell Culture with Intracellular Regulation

by Nikolay Bessonov, Gennady Bocharov, Anastasiia Mozokhina and Vitaly Volpert

Mathematics 2023, 11(6), 1526; https://0-doi-org.brum.beds.ac.uk/10.3390/math11061526 - 21 Mar 2023

Cited by 2 | Viewed by 1223

Abstract

Virus plaque assays are conventionally used for the assessment of viral infections, including their virulence, and vaccine efficacy. These experiments can be modeled with reaction–diffusion equations, allowing the estimation of the speed of infection spread (related to virus virulence) and viral load (related to virus infectivity). In this work, we develop a multiscale model of infection progression that combines macroscopic characterization of virus plaque growth in cell culture with a reference model of intracellular virus replication. We determine the infection spreading speed and viral load in a model for the extracellular dynamics and the kinetics of the abundance of intracellular viral genomes and proteins. In particular, the spatial infection spreading speed increases if the rate of virus entry into the target cell increases, while the viral load can either increase or decrease depending on other model parameters. The reduction in the model under a quasi-steady state assumption for some intracellular reactions allows us to derive a family of reduced models and to compare the reference model with the previous model for the concentration of uninfected cells, infected cells, and total virus concentration. Overall, the combination of different scales in reaction–diffusion models opens up new perspectives on virus plaque growth models and their applications. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

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47 pages, 5052 KiB

Open AccessArticle

Global Dynamics of a Diffusive Within-Host HTLV/HIV Co-Infection Model with Latency

by Noura H. AlShamrani, Ahmed Elaiw, Aeshah A. Raezah and Khalid Hattaf

Mathematics 2023, 11(6), 1523; https://0-doi-org.brum.beds.ac.uk/10.3390/math11061523 - 21 Mar 2023

Cited by 2 | Viewed by 1668

Abstract

In several publications, the dynamical system of HIV and HTLV mono-infections taking into account diffusion, as well as latently infected cells in cellular transmission has been mathematically analyzed. However, no work has been conducted on HTLV/HIV co-infection dynamics taking both factors into consideration. In this paper, a partial differential equations (PDEs) model of HTLV/HIV dual infection was developed and analyzed, considering the cells’ and viruses’ spatial mobility. CD

4^{+}

T cells are the primary target of both HTLV and HIV. For HIV, there are three routes of transmission: free-to-cell (FTC), latent infected-to-cell (ITC), and active ITC. In contrast, HTLV transmits horizontally through ITC contact and vertically through the mitosis of active HTLV-infected cells. In the beginning, the well-posedness of the model was investigated by proving the existence of global solutions and the boundedness. Eight threshold parameters that determine the existence and stability of the eight equilibria of the model were obtained. Lyapunov functions together with the Lyapunov–LaSalle asymptotic stability theorem were used to investigate the global stability of all equilibria. Finally, the theoretical results were verified utilizing numerical simulations. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

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33 pages, 1832 KiB

Open AccessArticle

Global Dynamics of an HTLV-I and SARS-CoV-2 Co-Infection Model with Diffusion

by Ahmed M. Elaiw, Abdulsalam S. Shflot, Aatef D. Hobiny and Shaban A. Aly

Mathematics 2023, 11(3), 688; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030688 - 29 Jan 2023

Cited by 1 | Viewed by 1335

Abstract

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is a novel respiratory virus that causes coronavirus disease 2019 (COVID-19). Symptoms of COVID-19 range from mild to severe illness. It was observed that disease progression in COVID-19 patients depends on their immune response, especially in elderly patients whose immune system suppression may put them at increased risk of infection. Human T-cell lymphotropic virus type-I (HTLV-I) attacks the CD4⁺ T cells (T cells) of the immune system and leads to immune dysfunction. Co-infection with HTLV-I and SARS-CoV-2 has been reported in recent studies. Modeling HTLV-I and SARS-CoV-2 co-infection can be a helpful tool to understand the in-host co-dynamics of these viruses. The aim of this study was to construct a model that characterizes the in-host dynamics of HTLV-I and SARS-CoV-2 co-infection. By considering the mobility of the viruses and cells, the model is represented by a system of partial differential equations (PDEs). The system contains two independent variables, time t and position x, and seven dependent variables for representing the densities of healthy epithelial cells (ECs), latent SARS-CoV-2-infected ECs, active SARS-CoV-2-infected ECs, SARS-CoV-2, healthy T cells, latent HTLV-I-infected T cells and active HTLV-I-infected T cells. We first studied the fundamental properties of the solutions of the system, then deduced all steady states and proved their global properties. We examined the global stability of the steady states by constructing appropriate Lyapunov functions. The analytical results were illustrated by performing numerical simulations. We discussed the effect of HTLV-I infection on COVID-19 progression. The results suggest that patients with HTLV-I have a weakened immune response; consequently, their risk of COVID-19 infection may be increased. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

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16 pages, 394 KiB

Open AccessArticle

Hopf Bifurcation in a Delayed Equation with Diffusion Driven by Carrying Capacity

by Yuanxian Hui, Yunfeng Liu and Zhong Zhao

Mathematics 2022, 10(14), 2382; https://0-doi-org.brum.beds.ac.uk/10.3390/math10142382 - 6 Jul 2022

Viewed by 1028

Abstract

In this paper, a delayed reaction–diffusion equation with carrying capacity-driven diffusion is investigated. The stability of the positive equilibrium solutions and the existence of the Hopf bifurcation of the equation are considered by studying the principal eigenvalue of an associated elliptic operator. The properties of the bifurcating periodic solutions are also obtained by using the normal form theory and the center manifold reduction. Furthermore, some representative numerical simulations are provided to illustrate the main theoretical results. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

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12 pages, 1153 KiB

Open AccessArticle

Uncertain Population Model with Jumps

by Caiwen Gao, Zhiqiang Zhang and Baoliang Liu

Mathematics 2022, 10(13), 2265; https://0-doi-org.brum.beds.ac.uk/10.3390/math10132265 - 28 Jun 2022

Viewed by 1234

Abstract

The uncertain population model (UPM), which has been proposed and studied, is a kind of population model driven by a Liu process that can only deal with continuous uncertain population systems. In reality, however, species systems may be suddenly shaken by earthquakes, tsunamis, epidemics, etc. The drastic changes lead to jumps in the population and make the sample path no longer continuous. In order to model the dramatic drifts embedded in an uncertain dynamic population system, this paper proposes a novel uncertain population model with jumps (UPMJ), which is described by a kind of uncertain differential equation with jumps (UDEJ). Then, the distribution function and the stability of solution for UPMJ are discussed based on uncertainty theory. Finally, a numerical example related to the transmission of Ebola virus is given to illustrate the characteristics of the distribution function and the stability of solution for UPMJ. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

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11 pages, 351 KiB

Open AccessArticle

Relaxation Oscillations and Dynamical Properties in a Time Delay Slow–Fast Predator–Prey Model with a Piecewise Smooth Functional Response

by Youhua Qian, Yuhui Peng, Yufeng Wang and Bingwen Lin

Mathematics 2022, 10(9), 1498; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091498 - 30 Apr 2022

Cited by 1 | Viewed by 1277

Abstract

In the past few decades, the predator–prey model has played an important role in the dynamic behavior of populations. Many scholars have studied the stability of the predator–prey system. Due to the complex influence of time delay on the dynamic behavior of systems, time-delay systems have garnered wide interest. In this paper, a classical piecewise smooth slow–fast predator–prey model is considered. The dynamic properties of the system are analyzed by linearization. The existence and uniqueness of the relaxation oscillation are then proven through the geometric singular perturbation theory and entry–exit function. Finally, a stable limit cycle is obtained. A numerical simulation verifies our results for the systems and shows the effectiveness of the method in dealing with time delays. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

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23 pages, 4840 KiB

Open AccessArticle

Global Dynamics of a Predator–Prey Model with Fear Effect and Impulsive State Feedback Control

by Yangyang Su and Tongqian Zhang

Mathematics 2022, 10(8), 1229; https://0-doi-org.brum.beds.ac.uk/10.3390/math10081229 - 8 Apr 2022

Cited by 1 | Viewed by 1669

Abstract

In this paper, a predator–prey model with fear effect and impulsive state control is proposed and analyzed. By constructing an appropriate Poincaré map, the dynamic properties of the system, including the existence, nonexistence, and stability of periodic solutions are studied. More specifically, based on the biological meaning, the pulse and the phase set are firstly defined in different regions as well as the corresponding Poincaré map. Subsequently, the properties of the Poincaré map are analyzed, and the existence of a periodic solution for the system is investigated according to the properties of the Poincaré map. We found that the existence of the periodic solution for the system completely depends on the property of the Poincaré map. Finally, several examples containing numerical simulations verify the obtained theoretical result. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

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10 pages, 760 KiB

Open AccessArticle

Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators

by Chun Zhang, Qiaoxia Tang and Zhixiang Wang

Mathematics 2022, 10(5), 772; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050772 - 28 Feb 2022

Viewed by 1383

Abstract

By introducing a switching scheme related to the state and time, a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system. Shooting methods to locate the limit cycle and specify bifurcation sets are described by defining an appropriate Poincaré map. Different types of multiple-Focus/Cycle and single-Focus/Cycle period oscillations in the system can be observed. Symmetry-breaking, period-doubling, and grazing bifurcation curves are obtained in the plane of bifurcation parameters, dividing the parameters plane into several regions corresponding to different kinds of oscillations. Meanwhile, based on the numerical simulation and bifurcation analysis, the mechanisms of several typical dynamical behaviors observed in different regions are presented. Full article

(This article belongs to the Special Issue Applications of Differential Equations to Mathematical Biology)

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