Control Theory and Its Application in Mathematical Biology

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 10257

Special Issue Editors


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Guest Editor
Asansol Girls' College, Kazi Nazrul University, Kolkata WB-713304, India
Interests: mathematical biology and ecology; ecoepidemiology; mathematical modeling; time-delayed systems; optimal control; impulsive control; dynamical systems; vector-borne plant and human diseases; awareness-based control measures; chemical and biochemical systems; infectious disease modeling and control; stability and bifurcation analysis

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Guest Editor
Department of Mathematics, University of Sussex, Falmer, Brighton BN1 9QH, UK
Interests: plant immune response and RNA interference; control of plant parasites; interactions between epidemics and awareness; dynamics of diseases on networks; pathogen-induced autoimmunity; stochastic effects in the dynamics of immune response; stochastic delayed models; time-delayed feedback control; stability and synchronization in networks with time-delayed connections; distributed time delays; numerical bifurcation analysis and simulations

Special Issue Information

Dear Colleagues, 

In recent decades, the optimal control theory has seen major advances, both in terms of theory and in its range of applications in various fields of science, engineering and technology. 

Optimal control problems quite often arise in mathematical models represented by systems of differential equations, with dynamical system techniques providing important insights into dynamics. Since these mathematical models represent real systems that arise in physics, chemistry and biology, one can either use existing control theory methods to achieve a desired outcome, while others may require the development of novel methods. One commonly used, popular approach for this context is that of feedback control, which can be instantaneous or delayed, depending on the specific problem at hand. 

This Special Issue aims to create a multidisciplinary collection of research articles concerning recent advances in the control theory and novel applications in biology, ecology, engineering and medicine. The main aim of this Special Issue is to highlight the latest advances in mathematical methods and procedures applied in the control theory. 

Potential topics include, but are not limited to, the following: 

  • Advances in control theory and applications;
  • Modeling and optimal control in biological systems;
  • Numerical methods for optimal control problems;
  • Optimal control of human and plant diseases;
  • Impulsive control approach;
  • Optimal impulsive control;
  • Optimal control theory in oncology;
  • The stochastic optimal control problem;
  • Mathematical modeling and control of industrial processes;
  • Mathematical modeling and control of energy systems;
  • Biomedical system modeling and control.

Dr. Fahad Al Basir
Dr. Konstantin Blyuss
Guest Editors

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 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

  • mathematical modeling and control
  • control theory and applications
  • optimal control approach
  • impulsive control approach
  • optimal impulsive control
  • process optimization
  • cost-effective analysis
  • time-delayed feedback control

Published Papers (6 papers)

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Research

18 pages, 367 KiB  
Article
Optimal Treatment Strategy for Cancer Based on Mathematical Modeling and Impulse Control Theory
by Wenhui Luo, Xuewen Tan and Juan Shen
Axioms 2023, 12(10), 916; https://doi.org/10.3390/axioms12100916 - 27 Sep 2023
Viewed by 853
Abstract
Adaptive therapy is a new type of cancer treatment in which time and dose are dynamically changed according to different individuals, which is very different from conventional cancer treatment strategies that use the maximum dose to kill the tumor cells. However, how to [...] Read more.
Adaptive therapy is a new type of cancer treatment in which time and dose are dynamically changed according to different individuals, which is very different from conventional cancer treatment strategies that use the maximum dose to kill the tumor cells. However, how to determine the time and dose of drug treatment is a challenging problem. In this paper, a competition model between drug-sensitive cells and drug-resistant cells was established, in which pulse intervention was introduced. In addition, based on the theory of pulse optimal control, three pulse optimal control strategies are proposed in the process of cancer treatment by controlling the pulse interval and dose, minimizing the number of tumor cells at the end of the day at minimal cost. Finally, three optimization strategies were compared, using numerical simulation, in terms of tumor burden and the effect on drug-resistant cells. The results show that the hybrid control strategy has the best effect. This work would provide some new ideas for the treatment of cancer. Full article
(This article belongs to the Special Issue Control Theory and Its Application in Mathematical Biology)
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30 pages, 680 KiB  
Article
The Optimal Strategies to Be Adopted in Controlling the Co-Circulation of COVID-19, Dengue and HIV: Insight from a Mathematical Model
by Andrew Omame, Aeshah A. Raezah, Uchenna H. Diala and Chinyere Onuoha
Axioms 2023, 12(8), 773; https://doi.org/10.3390/axioms12080773 - 9 Aug 2023
Cited by 10 | Viewed by 1002
Abstract
The pandemic caused by COVID-19 led to serious disruptions in the preventive efforts against other infectious diseases. In this work, a robust mathematical co-dynamical model of COVID-19, dengue, and HIV is designed. Rigorous analyses for investigating the dynamical properties of the designed model [...] Read more.
The pandemic caused by COVID-19 led to serious disruptions in the preventive efforts against other infectious diseases. In this work, a robust mathematical co-dynamical model of COVID-19, dengue, and HIV is designed. Rigorous analyses for investigating the dynamical properties of the designed model are implemented. Under a special case, the stability of the model’s equilibria is demonstrated using well-known candidates for the Lyapunov function. To reduce the co-circulation of the three diseases, optimal interventions were defined for the model and the control system was analyzed. Simulations of the model showed different control scenarios, which could have a positive or detrimental impact on reducing the co-circulation of the diseases. Highlights of the simulations included: (i) Upon implementation of the first intervention strategy (control against COVID-19 and dengue), it was observed that a significant number of single and dual infection cases were averted. (ii) Under the COVID-19 and HIV prevention strategy, a remarkable number of new single and dual infection cases were also prevented. (iii) Under the COVID-19 and co-infection prevention strategy, a significant number of new infections were averted. (iv) Comparing all the intervention measures considered in this study, it is possible to state that the strategy that combined COVID-19/HIV averted the highest number of new infections. Thus, the COVID-19/HIV strategy would be the ideal and optimal strategy to adopt in controlling the co-spread of COVID-19, dengue, and HIV. Full article
(This article belongs to the Special Issue Control Theory and Its Application in Mathematical Biology)
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28 pages, 991 KiB  
Article
Stability of HIV-1 Dynamics Models with Viral and Cellular Infections in the Presence of Macrophages
by Aeshah A. Raezah, Elsayed Dahy, E. Kh. Elnahary and Shaimaa A. Azoz
Axioms 2023, 12(7), 617; https://doi.org/10.3390/axioms12070617 - 21 Jun 2023
Viewed by 788
Abstract
In this research work, we suggest two mathematical models that take into account (i) two categories of target cells, CD4+T cells and macrophages, and (ii) two modes of infection transmissions, the direct virus-to-cell (VTC) method and cell-to-cell (CTC) infection transmission, [...] Read more.
In this research work, we suggest two mathematical models that take into account (i) two categories of target cells, CD4+T cells and macrophages, and (ii) two modes of infection transmissions, the direct virus-to-cell (VTC) method and cell-to-cell (CTC) infection transmission, where CTC is an effective method of spreading human immunodeficiency virus type-1 (HIV-1), as with the VTC method. The second model incorporates four time delays. In both models, the presence of a bounded and positive solution of the biological model is investigated. The existence conditions of all equilibria are established. The basic reproduction number R0 that identifies a disease index is obtained. Lyapunov functions are utilized to verify the global stability of all equilibria. The theoretical findings are verified through numerical simulations. According to the outcomes, the trajectories of the solutions approach the infection-free equilibrium and infection-present equilibrium when R01 and R0>1, respectively. Further, we study the sensitivity analysis to investigate how the values of all the parameters of the suggested model affect R0 for given data. We discuss the impact of the time delay on HIV-1 progression. We find that a longer time delay results in suppression of the HIV-1 infection and vice versa. Full article
(This article belongs to the Special Issue Control Theory and Its Application in Mathematical Biology)
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18 pages, 587 KiB  
Article
Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment
by Fahad Al Basir, Biru Rajak, Bootan Rahman and Khalid Hattaf
Axioms 2023, 12(6), 608; https://doi.org/10.3390/axioms12060608 - 19 Jun 2023
Cited by 1 | Viewed by 1219
Abstract
Infectious diseases continue to be a significant threat to human health and civilization, and finding effective methods to combat them is crucial. In this paper, we investigate the impact of awareness campaigns and optimal control techniques on infectious diseases without proper vaccines. Specifically, [...] Read more.
Infectious diseases continue to be a significant threat to human health and civilization, and finding effective methods to combat them is crucial. In this paper, we investigate the impact of awareness campaigns and optimal control techniques on infectious diseases without proper vaccines. Specifically, we develop an SIRS-type mathematical model that incorporates awareness campaigns through media and treatment for disease transmission dynamics and control. The model displays two equilibria, a disease-free equilibrium and an endemic equilibrium, and exhibits Hopf bifurcation when the bifurcation parameter exceeds its critical value, causing a switch in the stability of the system. We also propose an optimal control problem that minimizes the cost of control measures while achieving a desired level of disease control. By applying the minimum principle to the optimal control problem, we obtain analytical and numerical results that show how the infection rate of the disease affects the stability of the system and how awareness campaigns and treatment can maintain the stability of the system. This study highlights the importance of awareness campaigns in controlling infectious diseases and demonstrates the effectiveness of optimal control theory in achieving disease control with minimal cost. Full article
(This article belongs to the Special Issue Control Theory and Its Application in Mathematical Biology)
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12 pages, 349 KiB  
Article
A Numerical Implementation of Fractional-Order PID Controllers for Autonomous Vehicles
by Iqbal M. Batiha, Osama Y. Ababneh, Abeer A. Al-Nana, Waseem G. Alshanti, Shameseddin Alshorm and Shaher Momani
Axioms 2023, 12(3), 306; https://doi.org/10.3390/axioms12030306 - 17 Mar 2023
Cited by 8 | Viewed by 1627
Abstract
In the context of reaching the best way to control the movement of autonomous cars linearly and angularly, making them more stable and balanced on different roads and ensuring that they avoid road obstacles, this manuscript chiefly aims to reach the optimal approach [...] Read more.
In the context of reaching the best way to control the movement of autonomous cars linearly and angularly, making them more stable and balanced on different roads and ensuring that they avoid road obstacles, this manuscript chiefly aims to reach the optimal approach for a fractional-order PID controller (or PIγDρ-controller) instead of the already classical one used to provide smooth automatic parking for electrical autonomous cars. The fractional-order PIγDρ-controller is based on the particle swarm optimization (PSO) algorithm for its design, with two different approximations: Oustaloup’s approximation and the continued fractional expansion (CFE) approximation. Our approaches to the fractional-order PID using the results of the PSO algorithm are compared with the classical PID that was designed using the results of the Cohen–Coon, Ziegler–Nichols and bacteria foraging algorithms. The scheme represented by the proposed PIγDρ-controller can provide the system of the autonomous vehicle with more stable results than that of the PID controller. Full article
(This article belongs to the Special Issue Control Theory and Its Application in Mathematical Biology)
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26 pages, 2331 KiB  
Article
Dynamical Properties of Discrete-Time HTLV-I and HIV-1 within-Host Coinfection Model
by Ahmed M. Elaiw, Abdulaziz K. Aljahdali and Aatef D. Hobiny
Axioms 2023, 12(2), 201; https://doi.org/10.3390/axioms12020201 - 14 Feb 2023
Cited by 5 | Viewed by 3665
Abstract
Infection with human immunodeficiency virus type 1 (HIV-1) or human T-lymphotropic virus type I (HTLV-I) or both can lead to mortality. CD4+T cells are the target for both HTLV-I and HIV-1. In addition, HIV-1 can infect macrophages. CD4+T cells [...] Read more.
Infection with human immunodeficiency virus type 1 (HIV-1) or human T-lymphotropic virus type I (HTLV-I) or both can lead to mortality. CD4+T cells are the target for both HTLV-I and HIV-1. In addition, HIV-1 can infect macrophages. CD4+T cells and macrophages play important roles in the immune system response. This article develops and analyzes a discrete-time HTLV-I and HIV-1 co-infection model. The model depicts the within-host interaction of six compartments: uninfected CD4+T cells, HIV-1-infected CD4+T cells, uninfected macrophages, HIV-1-infected macrophages, free HIV-1 particles and HTLV-I-infected CD4+T cells. The discrete-time model is obtained by discretizing the continuous-time model via the nonstandard finite difference (NSFD) approach. We show that NSFD preserves the positivity and boundedness of the model’s solutions. We deduce four threshold parameters that control the existence and stability of the four equilibria of the model. The Lyapunov method is used to examine the global stability of all equilibria. The analytical findings are supported via numerical simulation. The model can be useful when one seeks to design optimal treatment schedules using optimal control theory. Full article
(This article belongs to the Special Issue Control Theory and Its Application in Mathematical Biology)
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