1
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Enatsu Y, Roy J, Banerjee M. Hunting cooperation in a prey-predator model with maturation delay. J Biol Dyn 2024; 18:2332279. [PMID: 38517146 DOI: 10.1080/17513758.2024.2332279] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2023] [Accepted: 03/12/2024] [Indexed: 03/23/2024]
Abstract
We investigate the dynamics of a prey-predator model with cooperative hunting among specialist predators and maturation delay in predator growth. First, we consider a model without delay and explore the effect of hunting time on the coexistence of predator and their prey. When the hunting time is long enough and the cooperation rate among predators is weak, prey and predator species tend to coexist. Furthermore, we observe the occurrences of a series of bifurcations that depend on the cooperation rate and the hunting time. Second, we introduce a maturation delay for predator growth and analyse its impact on the system's dynamics. We find that as the delay becomes larger, predator species become more likely to go extinct, as the long maturation delay hinders the growth of the predator population. Our numerical exploration reveals that the delay causes shifts in both the bifurcation curves and bifurcation thresholds of the non-delayed system.
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Affiliation(s)
- Yoichi Enatsu
- Institute of Arts and Sciences, Oshamambe Division, Tokyo University of Science, Hokkaido, Japan
| | - Jyotirmoy Roy
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India
- Department of Basic Science and Humanities, University of Engineering and Management, Kolkata, India
| | - Malay Banerjee
- Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India
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2
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Dai Q, Zhou J, Kong Z. The role of directed cycles in a directed neural network. Neural Netw 2024; 176:106329. [PMID: 38653127 DOI: 10.1016/j.neunet.2024.106329] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2023] [Revised: 03/09/2024] [Accepted: 04/17/2024] [Indexed: 04/25/2024]
Abstract
This paper investigates the dynamics of a directed acyclic neural network by edge adding control. We find that the local stability and Hopf bifurcation of the controlled network only depend on the size and intersection of directed cycles, instead of the number and position of the added edges. More specifically, if there is no cycle in the controlled network, the local dynamics of the network will remain unchanged and Hopf bifurcation will not occur even if the number of added edges is sufficient. However, if there exist cycles, then the network may undergo Hopf bifurcation. Our results show that the cycle structure is a necessary condition for the generation of Hopf bifurcation, and the bifurcation threshold is determined by the number, size, and intersection of cycles. Numerical experiments are provided to support the validity of the theory.
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Affiliation(s)
- Qinrui Dai
- School of Mathematics and Statistics, Wuhan University, Hubei 430072, China
| | - Jin Zhou
- School of Mathematics and Statistics, Wuhan University, Hubei 430072, China; Hubei Key Laboratory of Computational Science, Wuhan University, Hubei 430072, China.
| | - Zhengmin Kong
- School of Electrical Engineering and Automation, Wuhan University, Hubei 430072, China
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3
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Xu C, Lin J, Zhao Y, Cui Q, Ou W, Pang Y, Liu Z, Liao M, Li P. New results on bifurcation for fractional-order octonion-valued neural networks involving delays. Network 2024:1-53. [PMID: 38578214 DOI: 10.1080/0954898x.2024.2332662] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2023] [Accepted: 03/09/2024] [Indexed: 04/06/2024]
Abstract
This work chiefly explores fractional-order octonion-valued neural networks involving delays. We decompose the considered fractional-order delayed octonion-valued neural networks into equivalent real-valued systems via Cayley-Dickson construction. By virtue of Lipschitz condition, we prove that the solution of the considered fractional-order delayed octonion-valued neural networks exists and is unique. By constructing a fairish function, we confirm that the solution of the involved fractional-order delayed octonion-valued neural networks is bounded. Applying the stability theory and basic bifurcation knowledge of fractional order differential equations, we set up a sufficient condition remaining the stability behaviour and the appearance of Hopf bifurcation for the addressed fractional-order delayed octonion-valued neural networks. To illustrate the justifiability of the derived theoretical results clearly, we give the related simulation results to support these facts. Simultaneously, the bifurcation plots are also displayed. The established theoretical results in this work have important guiding significance in devising and improving neural networks.
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Affiliation(s)
- Changjin Xu
- Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Jinting Lin
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Yingyan Zhao
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Qingyi Cui
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Wei Ou
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Yicheng Pang
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Zixin Liu
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Maoxin Liao
- School of Mathematics and Physics, University of South China, Hengyang, P.R. China
| | - Peiluan Li
- School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, P.R. China
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4
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Wu SX, Wang ZC, Ruan S. Hopf bifurcation in an age-structured predator-prey system with Beddington-DeAngelis functional response and constant harvesting. J Math Biol 2024; 88:56. [PMID: 38573424 PMCID: PMC10995042 DOI: 10.1007/s00285-024-02070-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/22/2023] [Revised: 11/26/2023] [Accepted: 02/29/2024] [Indexed: 04/05/2024]
Abstract
In this paper, an age-structured predator-prey system with Beddington-DeAngelis (B-D) type functional response, prey refuge and harvesting is investigated, where the predator fertility function f(a) and the maturation function β ( a ) are assumed to be piecewise functions related to their maturation period τ . Firstly, we rewrite the original system as a non-densely defined abstract Cauchy problem and show the existence of solutions. In particular, we discuss the existence and uniqueness of a positive equilibrium of the system. Secondly, we consider the maturation period τ as a bifurcation parameter and show the existence of Hopf bifurcation at the positive equilibrium by applying the integrated semigroup theory and Hopf bifurcation theorem. Moreover, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied by applying the center manifold theorem and normal form theory. Finally, some numerical simulations are given to illustrate of the theoretical results and a brief discussion is presented.
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Affiliation(s)
- San-Xing Wu
- School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People's Republic of China
| | - Zhi-Cheng Wang
- School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, Gansu, People's Republic of China
| | - Shigui Ruan
- Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA.
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5
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Bugalia S, Tripathi JP, Wang H. Mutations make pandemics worse or better: modeling SARS-CoV-2 variants and imperfect vaccination. J Math Biol 2024; 88:45. [PMID: 38507066 DOI: 10.1007/s00285-024-02068-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/30/2021] [Revised: 07/04/2023] [Accepted: 02/18/2024] [Indexed: 03/22/2024]
Abstract
COVID-19 is a respiratory disease triggered by an RNA virus inclined to mutations. Since December 2020, variants of COVID-19 (especially Delta and Omicron) continuously appeared with different characteristics that influenced death and transmissibility emerged around the world. To address the novel dynamics of the disease, we propose and analyze a dynamical model of two strains, namely native and mutant, transmission dynamics with mutation and imperfect vaccination. It is also assumed that the recuperated individuals from the native strain can be infected with mutant strain through the direct contact with individual or contaminated surfaces or aerosols. We compute the basic reproduction number, R 0 , which is the maximum of the basic reproduction numbers of native and mutant strains. We prove the nonexistence of backward bifurcation using the center manifold theory, and global stability of disease-free equilibrium whenR 0 < 1 , that is, vaccine is effective enough to eliminate the native and mutant strains even if it cannot provide full protection. Hopf bifurcation appears when the endemic equilibrium loses its stability. An intermediate mutation rate ν 1 leads to oscillations. When ν 1 increases over a threshold, the system regains its stability and exhibits an interesting dynamics called endemic bubble. An analytical expression for vaccine-induced herd immunity is derived. The epidemiological implication of the herd immunity threshold is that the disease may effectively be eradicated if the minimum herd immunity threshold is attained in the community. Furthermore, the model is parameterized using the Indian data of the cumulative number of confirmed cases and deaths of COVID-19 from March 1 to September 27 in 2021, using MCMC method. The cumulative cases and deaths can be reduced by increasing the vaccine efficacies to both native and mutant strains. We observe that by considering the vaccine efficacy against native strain as 90%, both cumulative cases and deaths would be reduced by 0.40%. It is concluded that increasing immunity against mutant strain is more influential than the vaccine efficacy against it in controlling the total cases. Our study demonstrates that the COVID-19 pandemic may be worse due to the occurrence of oscillations for certain mutation rates (i.e., outbreaks will occur repeatedly) but better due to stability at a lower infection level with a larger mutation rate. We perform sensitivity analysis using the Latin Hypercube Sampling methodology and partial rank correlation coefficients to illustrate the impact of parameters on the basic reproduction number, the number of cumulative cases and deaths, which ultimately sheds light on disease mitigation.
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Affiliation(s)
- Sarita Bugalia
- Department of Mathematics, Central University of Rajasthan, Bandar Sindri, Kishangarh, Ajmer, Rajasthan, 305817, India
| | - Jai Prakash Tripathi
- Department of Mathematics, Central University of Rajasthan, Bandar Sindri, Kishangarh, Ajmer, Rajasthan, 305817, India.
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada
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Pan Q, Lu M, Huang J, Ruan S. Effects of whaling and krill fishing on the whale-krill predation dynamics: bifurcations in a harvested predator-prey model with Holling type I functional response. J Math Biol 2024; 88:42. [PMID: 38446242 DOI: 10.1007/s00285-024-02063-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/23/2023] [Revised: 02/05/2024] [Accepted: 02/11/2024] [Indexed: 03/07/2024]
Abstract
In the Antarctic, the whale population had been reduced dramatically due to the unregulated whaling. It was expected that Antarctic krill, the main prey of whales, would grow significantly as a consequence and exploratory krill fishing was practiced in some areas. However, it was found that there has been a substantial decline in abundance of krill since the end of whaling, which is the phenomenon of krill paradox. In this paper, to study the krill-whale interaction we revisit a harvested predator-prey model with Holling I functional response. We find that the model admits at most two positive equilibria. When the two positive equilibria are located in the region{ ( N , P ) | 0 ≤ N < 2 N c , P ≥ 0 } , the model exhibits degenerate Bogdanov-Takens bifurcation with codimension up to 3 and Hopf bifurcation with codimension up to 2 by rigorous bifurcation analysis. When the two positive equilibria are located in the region{ ( N , P ) | N > 2 N c , P ≥ 0 } , the model has no complex bifurcation phenomenon. When there is one positive equilibrium on each side of N = 2 N c , the model undergoes Hopf bifurcation with codimension up to 2. Moreover, numerical simulation reveals that the model not only can exhibit the krill paradox phenomenon but also has three limit cycles, with the outmost one crosses the line N = 2 N c under some specific parameter conditions.
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Affiliation(s)
- Qin Pan
- School of Mathematics and Statistics, and Key Lab NAA-MOE, Central China Normal University, Wuhan, 430079, China
| | - Min Lu
- School of Mathematics and Statistics, and Key Lab NAA-MOE, Central China Normal University, Wuhan, 430079, China.
| | - Jicai Huang
- School of Mathematics and Statistics, and Key Lab NAA-MOE, Central China Normal University, Wuhan, 430079, China
| | - Shigui Ruan
- Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA
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7
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Essongo FE, Mvogo A, Ben-Bolie GH. Dynamics of a diffusive model for cancer stem cells with time delay in microRNA-differentiated cancer cell interactions and radiotherapy effects. Sci Rep 2024; 14:5295. [PMID: 38438408 PMCID: PMC10912232 DOI: 10.1038/s41598-024-55212-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/14/2023] [Accepted: 02/21/2024] [Indexed: 03/06/2024] Open
Abstract
Understand the dynamics of cancer stem cells (CSCs), prevent the non-recurrence of cancers and develop therapeutic strategies to destroy both cancer cells and CSCs remain a challenge topic. In this paper, we study both analytically and numerically the dynamics of CSCs under radiotherapy effects. The dynamical model takes into account the diffusion of cells, the de-differentiation (or plasticity) mechanism of differentiated cancer cells (DCs) and the time delay on the interaction between microRNAs molecules (microRNAs) with DCs. The stability of the model system is studied by using a Hopf bifurcation analysis. We mainly investigate on the critical time delay τ c , that represents the time for DCs to transform into CSCs after the interaction of microRNAs with DCs. Using the system parameters, we calculate the value of τ c for prostate, lung and breast cancers. To confirm the analytical predictions, the numerical simulations are performed and show the formation of spatiotemporal circular patterns. Such patterns have been found as promising diagnostic and therapeutic value in management of cancer and various diseases. The radiotherapy is applied in the particular case of prostate model. We calculate the optimum dose of radiation and determine the probability of avoiding local cancer recurrence after radiotherapy treatment. We find numerically a complete eradication of patterns when the radiotherapy is applied before a time t < τ c . This scenario induces microRNAs to act as suppressors as experimentally observed in prostate cancer. The results obtained in this paper will provide a better concept for the clinicians and oncologists to understand the complex dynamics of CSCs and to design more efficacious therapeutic strategies to prevent the non-recurrence of cancers.
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Affiliation(s)
- Frank Eric Essongo
- Laboratory of Nuclear Physics, Dosimetry and Radiation Protection, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon
| | - Alain Mvogo
- Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon.
| | - Germain Hubert Ben-Bolie
- Laboratory of Nuclear Physics, Dosimetry and Radiation Protection, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon
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8
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Guo Q, Wang L, Liu H, Wang Y, Li J, Kumar Tiwari P, Zhao M, Dai C. Stability switches and chaos induced by delay in a reaction-diffusion nutrient-plankton model. J Biol Dyn 2023; 17:2272852. [PMID: 37962904 DOI: 10.1080/17513758.2023.2272852] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2022] [Accepted: 10/14/2023] [Indexed: 11/15/2023]
Abstract
In this paper, we investigate a reaction-diffusion model incorporating dynamic variables for nutrient, phytoplankton, and zooplankton. Moreover, we account for the impact of time delay in the growth of phytoplankton following nutrient uptake. Our theoretical analysis reveals that the time delay can trigger the emergence of persistent oscillations in the model via a Hopf bifurcation. We also analytically track the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Our simulation results demonstrate stability switches occurring for the positive equilibrium with an increasing time lag. Furthermore, the model exhibits homogeneous periodic-2 and 3 solutions, as well as chaotic behaviour. These findings highlight that the presence of time delay in the phytoplankton growth can bring forth dynamical complexity to the nutrient-plankton system of aquatic habitats.
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Affiliation(s)
- Qing Guo
- School of Life and Environmental Science, Wenzhou University, Wenzhou, People's Republic of China
- Environmental Engineering Program University of Northern British Columbia, Prince George, Canada
| | - Lijun Wang
- School of Intelligent Manufacturing and Vehicle Engineering, Sichuan Institute of Industrial Technology, Deyang, People's Republic of China
| | - He Liu
- School of Life and Environmental Science, Wenzhou University, Wenzhou, People's Republic of China
- Environmental Engineering Program University of Northern British Columbia, Prince George, Canada
| | - Yi Wang
- School of Life and Environmental Science, Wenzhou University, Wenzhou, People's Republic of China
| | - Jianbing Li
- Environmental Engineering Program University of Northern British Columbia, Prince George, Canada
- WZU-UNBC Joint Research Institute of Ecology and Environment, Wenzhou University, Wenzhou, People's Republic of China
| | - Pankaj Kumar Tiwari
- Department of Basic Science and Humanities, Indian Institute of Information Technology, Bhagalpur, India
| | - Min Zhao
- School of Life and Environmental Science, Wenzhou University, Wenzhou, People's Republic of China
- National & Local Joint Engineering Research Center for Ecological Treatment Technology of Urban Water Pollution, Wenzhou University, Wenzhou, People's Republic of China
| | - Chuanjun Dai
- School of Life and Environmental Science, Wenzhou University, Wenzhou, People's Republic of China
- National & Local Joint Engineering Research Center for Ecological Treatment Technology of Urban Water Pollution, Wenzhou University, Wenzhou, People's Republic of China
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9
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Zhu Z, Chen Y, Chen F, Li Z. Complex dynamics of a predator-prey model with opportunistic predator and weak Allee effect in prey. J Biol Dyn 2023; 17:2225545. [PMID: 37339327 DOI: 10.1080/17513758.2023.2225545] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2022] [Accepted: 06/10/2023] [Indexed: 06/22/2023]
Abstract
In this work, we first modify a Lotka-Volterra predator-prey system to incorporate an opportunistic predator and weak Allee effect in prey. The prey will be extinct if the combined effect of hunting and other food resources of predator is large. Otherwise, the dynamic behaviour of the system is extremely rich. A series of bifurcations such as saddle-node bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation can happen. The validity of the theoretical results are supported with numerical simulations.
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Affiliation(s)
- Zhenliang Zhu
- College of Mathematics and Data Science, Minjiang University, Fuzhou, Fujian, People's Republic of China
| | - Yuming Chen
- Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada
| | - Fengde Chen
- College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian, People's Republic of China
| | - Zhong Li
- College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian, People's Republic of China
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10
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Fung T, Clapham HE, Chisholm RA. Temporary Cross-Immunity as a Plausible Driver of Asynchronous Cycles of Dengue Serotypes. Bull Math Biol 2023; 85:124. [PMID: 37962713 DOI: 10.1007/s11538-023-01226-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2023] [Accepted: 10/16/2023] [Indexed: 11/15/2023]
Abstract
Many infectious diseases exist as multiple variants, with interactions between variants potentially driving epidemiological dynamics. These diseases include dengue, which infects hundreds of millions of people every year and exhibits complex multi-serotype dynamics. Antibodies produced in response to primary infection by one of the four dengue serotypes can produce a period of temporary cross-immunity (TCI) to infection by other serotypes. After this period, the remaining antibodies can facilitate the entry of heterologous serotypes into target cells, thus enhancing severity of secondary infection by a heterologous serotype. This represents antibody-dependent enhancement (ADE). In this study, we analyze an epidemiological model to provide novel insights into the importance of TCI and ADE in producing cyclic outbreaks of dengue serotypes. Our analyses reveal that without TCI, such cyclic outbreaks are synchronous across serotypes and only occur when ADE produces high transmission rates. In contrast, the presence of TCI allows asynchronous cycles of serotypes by inducing a time lag between recovery from primary infection by one serotype and secondary infection by another, with such cycles able to occur without ADE. Our results suggest that TCI is a fundamental driver of asynchronous cycles of dengue serotypes and possibly other multi-variant diseases.
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Affiliation(s)
- Tak Fung
- Department of Biological Sciences, National University of Singapore, 16 Science Drive 4, Singapore, 117558, Singapore.
| | - Hannah E Clapham
- Saw Swee Hock School of Public Health, National University of Singapore and National University Health System, 12 Science Drive 2, Singapore, 117549, Singapore
| | - Ryan A Chisholm
- Department of Biological Sciences, National University of Singapore, 16 Science Drive 4, Singapore, 117558, Singapore
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11
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Huang C, Wang H, Liu H, Cao J. Bifurcations of a delayed fractional-order BAM neural network via new parameter perturbations. Neural Netw 2023; 168:123-142. [PMID: 37748392 DOI: 10.1016/j.neunet.2023.08.060] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/09/2023] [Revised: 05/30/2023] [Accepted: 08/31/2023] [Indexed: 09/27/2023]
Abstract
This paper makes a new breakthrough in deliberating the bifurcations of fractional-order bidirectional associative memory neural network (FOBAMNN). In the beginning, the corresponding bifurcation results are established according to self-regulating parameter, which is different from bifurcation outcomes available by using time delay as the bifurcation parameter, and greatly enriches the bifurcation results of continuous neural networks(NNs). The deived results manifest that a larger self-regulating parameter is more conducive to the stability of the system, which is consistent with the actual meaning of the self-regulating parameter representing the decay rate of activity. In addition to the innovation in the research object, this paper also has innovation in the procedure of calculating the bifurcation critical point. In the face of the quartic equation about the bifurcation parameters, this paper utilizes the methodology of implicit array to calculate the bifurcation critical point succinctly and effectively, which eschews the disadvantages of the conventional Ferrari approach, such as cumbersome formula and huge computational efforts. Our developed technique can be employed as a general method to solve the bifurcation point including the problem of dealing with the bifurcation critical point of delay. Ultimately, numerical experiments test the key theoretical fruits of this paper.
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Affiliation(s)
- Chengdai Huang
- School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China.
| | - Huanan Wang
- School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China.
| | - Heng Liu
- School of Mathematics and Physics, Guangxi Minzu University, Nanning 530006, China.
| | - Jinde Cao
- School of Mathematics, Southeast University, Nanjing 210096, China; Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea.
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12
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Rao F, Zhang Z, Li J. Dynamical analysis of a glucose-insulin regulatory system with insulin-degrading enzyme and multiple delays. J Math Biol 2023; 87:73. [PMID: 37856001 DOI: 10.1007/s00285-023-02003-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/02/2021] [Revised: 12/28/2022] [Accepted: 09/18/2023] [Indexed: 10/20/2023]
Abstract
This paper investigates the dynamics of a glucose-insulin regulatory system model that incorporates: (1) insulin-degrading enzyme in the insulin equation; and (2) discrete time delays respectively in the insulin production term, hepatic glucose production term, and the insulin-degrading enzyme. We provide rigorous results of our model including the asymptotic stability of the equilibrium solution and the existence of Hopf bifurcation. We show that analytically and numerically at a certain value the time delays driven stability or instability occurs when the corresponding model has an interior equilibrium. Moreover, we illustrate the oscillatory regulation and insulin secretion via numerical simulations, which show that the model dynamics exhibit physiological observations and more information by allowing parameters to vary. Our results may provide useful biological insights into diabetes for the glucose-insulin regulatory system model.
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Affiliation(s)
- Feng Rao
- School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, 211816, Jiangsu, China.
| | - Zhongliang Zhang
- School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, 211816, Jiangsu, China
| | - Jiaxu Li
- Department of Mathematics, University of Louisville, Louisville, KY, 40292, USA.
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13
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Yang J, Jia P, Wang J, Jin Z. Rich dynamics of a bidirectionally linked immuno-epidemiological model for cholera. J Math Biol 2023; 87:71. [PMID: 37848667 DOI: 10.1007/s00285-023-02009-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2023] [Revised: 09/24/2023] [Accepted: 10/03/2023] [Indexed: 10/19/2023]
Abstract
Cholera is an environmentally driven disease where the human hosts both ingest the pathogen from polluted environment and shed the pathogen to the environment, generating a two-way feedback cycle. In this paper, we propose a bidirectionally linked immuno-epidemiological model to study the interaction of within- and between-host cholera dynamics. We conduct a rigorous analysis for this multiscale model, with a focus on the stability and bifurcation properties of each feasible equilibrium. We find that the parameter that represents the bidirectional connection is a key factor in shaping the rich dynamics of the system, including the occurrence of the backward bifurcation and Hopf bifurcation. Numerical results illustrate a practical application of our model and add new insight into the prevention and intervention of cholera epidemics.
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Affiliation(s)
- Junyuan Yang
- Complex Systems Research Center, Shanxi University, Taiyuan, 030006, Shanxi, People's Republic of China.
- Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan, 030006, People's Republic of China.
| | - Peiqi Jia
- Complex Systems Research Center, Shanxi University, Taiyuan, 030006, Shanxi, People's Republic of China
- Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan, 030006, People's Republic of China
| | - Jin Wang
- Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN, 37403, USA
| | - Zhen Jin
- Complex Systems Research Center, Shanxi University, Taiyuan, 030006, Shanxi, People's Republic of China
- Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan, 030006, People's Republic of China
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14
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Blair DS, Soriano-Mas C, Cabral J, Moreira P, Morgado P, Deco G. Corrigendum: Complexity changes in functional state dynamics suggest focal connectivity reductions. Front Hum Neurosci 2023; 17:1275387. [PMID: 37886692 PMCID: PMC10599009 DOI: 10.3389/fnhum.2023.1275387] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2023] [Accepted: 09/29/2023] [Indexed: 10/28/2023] Open
Abstract
[This corrects the article DOI: 10.3389/fnhum.2022.958706.].
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Affiliation(s)
| | - Carles Soriano-Mas
- Psychiatry and Mental Health Group, Neuroscience Program, Institut d'Investigació Biomèdica de Bellvitge, Barcelona, Spain
- Network Center for Biomedical Research on Mental Health, Carlos III Health Institute, Madrid, Spain
- Department of Social Psychology and Quantitative Psychology, Universitat de Barcelona, Barcelona, Spain
| | - Joana Cabral
- Life and Health Sciences Research Institute, School of Medicine, University of Minho, Braga, Portugal
| | - Pedro Moreira
- Life and Health Sciences Research Institute, School of Medicine, University of Minho, Braga, Portugal
- ICVS/3B's, PT Government Associate Laboratory, Braga, Portugal
- Psychological Neuroscience Lab, CIPsi, School of Psychology, University of Minho, Braga, Portugal
| | - Pedro Morgado
- Life and Health Sciences Research Institute, School of Medicine, University of Minho, Braga, Portugal
- ICVS/3B's, PT Government Associate Laboratory, Braga, Portugal
- Clinical Academic Center—Braga, Braga, Portugal
| | - Gustavo Deco
- Facultad de Comunicación, Universitat Pompeu Fabra, Barcelona, Spain
- Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain
- Department of Neuropsychology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany
- School of Psychological Sciences, Monash University, Clayton, VIC, Australia
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15
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Abstract
Despite widespread and striking examples of physiological oscillations, their functional role is often unclear. Even glycolysis, the paradigm example of oscillatory biochemistry, has seen questions about its oscillatory function. Here, we take a systems approach to argue that oscillations play critical physiological roles, such as enabling systems to avoid desensitization, to avoid chronically high and therefore toxic levels of chemicals, and to become more resistant to noise. Oscillation also enables complex physiological systems to reconcile incompatible conditions such as oxidation and reduction, by cycling between them, and to synchronize the oscillations of many small units into one large effect. In pancreatic β-cells, glycolytic oscillations synchronize with calcium and mitochondrial oscillations to drive pulsatile insulin release, critical for liver regulation of glucose. In addition, oscillation can keep biological time, essential for embryonic development in promoting cell diversity and pattern formation. The functional importance of oscillatory processes requires a re-thinking of the traditional doctrine of homeostasis, holding that physiological quantities are maintained at constant equilibrium values, a view that has largely failed in the clinic. A more dynamic approach will initiate a paradigm shift in our view of health and disease. A deeper look into the mechanisms that create, sustain and abolish oscillatory processes requires the language of nonlinear dynamics, well beyond the linearization techniques of equilibrium control theory. Nonlinear dynamics enables us to identify oscillatory ('pacemaking') mechanisms at the cellular, tissue and system levels.
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Affiliation(s)
- Lingyun Ivy Xiong
- Department of Stem Cell Biology and Regenerative Medicine, Eli and Edythe Broad Center for Regenerative Medicine and Stem Cell Research, Keck School of Medicine of the University of Southern California, Los Angeles, CA, USA
- Department of Quantitative and Computational Biology, University of Southern California, Los Angeles, CA, USA
| | - Alan Garfinkel
- Departments of Medicine (Cardiology) and Integrative Biology and Physiology, University of California, Los Angeles, CA, USA
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16
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Wang C, Wang H, Yuan S. Precipitation governing vegetation patterns in an arid or semi-arid environment. J Math Biol 2023; 87:22. [PMID: 37395848 DOI: 10.1007/s00285-023-01954-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/28/2022] [Revised: 05/15/2023] [Accepted: 06/15/2023] [Indexed: 07/04/2023]
Abstract
In an arid or semi-arid environment, precipitation plays a vital role in vegetation growth. Recent researches reveal that the response of vegetation growth to precipitation has a lag effect. To explore the mechanism behind the lag phenomenon, we propose and investigate a water-vegetation model with spatiotemporal nonlocal effects. It is shown that the temporal kernel function does not affect Turing bifurcation. For better understanding the influences of lag effect and nonlocal competition on the vegetation pattern formation, we choose some special kernel functions and obtain some insightful results: (i) Time delay does not trigger the vegetation pattern formation, but can postpone the evolution of vegetation. In addition, in the absence of diffusion, time delay can induce the occurrence of stability switches, while in the presence of diffusion, spatially nonhomogeneous time-periodic solutions may emerge, but there are no stability switches; (ii) The spatial nonlocal interaction may trigger the pattern onset for small diffusion ratio of water and vegetation, and can change the number and size of isolated vegetation patches for large diffusion ratio. (iii) The interaction between time delay and spatial nonlocal competition may induce the emergence of traveling wave patterns, so that the vegetation remains periodic in space, but is oscillating in time. These results demonstrate that precipitation can significantly affect the growth and spatial distribution of vegetation.
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Affiliation(s)
- Cuihua Wang
- University of Shanghai for Science and Technology, Shanghai, 200093, People's Republic of China
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G2G1, Canada
| | - Sanling Yuan
- University of Shanghai for Science and Technology, Shanghai, 200093, People's Republic of China.
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17
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Perl YS, Zamora-Lopez G, Montbrió E, Monge-Asensio M, Vohryzek J, Fittipaldi S, Campo CG, Moguilner S, Ibañez A, Tagliazucchi E, Yeo BTT, Kringelbach ML, Deco G. The impact of regional heterogeneity in whole-brain dynamics in the presence of oscillations. Netw Neurosci 2023; 7:632-660. [PMID: 37397876 PMCID: PMC10312285 DOI: 10.1162/netn_a_00299] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2022] [Accepted: 12/02/2022] [Indexed: 12/25/2023] Open
Abstract
Large variability exists across brain regions in health and disease, considering their cellular and molecular composition, connectivity, and function. Large-scale whole-brain models comprising coupled brain regions provide insights into the underlying dynamics that shape complex patterns of spontaneous brain activity. In particular, biophysically grounded mean-field whole-brain models in the asynchronous regime were used to demonstrate the dynamical consequences of including regional variability. Nevertheless, the role of heterogeneities when brain dynamics are supported by synchronous oscillating state, which is a ubiquitous phenomenon in brain, remains poorly understood. Here, we implemented two models capable of presenting oscillatory behavior with different levels of abstraction: a phenomenological Stuart-Landau model and an exact mean-field model. The fit of these models informed by structural- to functional-weighted MRI signal (T1w/T2w) allowed us to explore the implication of the inclusion of heterogeneities for modeling resting-state fMRI recordings from healthy participants. We found that disease-specific regional functional heterogeneity imposed dynamical consequences within the oscillatory regime in fMRI recordings from neurodegeneration with specific impacts on brain atrophy/structure (Alzheimer's patients). Overall, we found that models with oscillations perform better when structural and functional regional heterogeneities are considered, showing that phenomenological and biophysical models behave similarly at the brink of the Hopf bifurcation.
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Affiliation(s)
- Yonatan Sanz Perl
- Department of Physics, University of Buenos Aires, Buenos Aires, Argentina
- National Scientific and Technical Research Council (CONICET), CABA, Buenos Aires, Argentina
- Cognitive Neuroscience Center (CNC), Universidad de San Andrés, Buenos Aires, Argentina
- Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Barcelona, Spain
| | - Gorka Zamora-Lopez
- Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Barcelona, Spain
| | - Ernest Montbrió
- Neuronal Dynamics Group, Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
| | - Martí Monge-Asensio
- Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Barcelona, Spain
| | - Jakub Vohryzek
- Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Barcelona, Spain
- Centre for Eudaimonia and Human Flourishing, University of Oxford, Oxford, United Kingdom
| | - Sol Fittipaldi
- National Scientific and Technical Research Council (CONICET), CABA, Buenos Aires, Argentina
- Cognitive Neuroscience Center (CNC), Universidad de San Andrés, Buenos Aires, Argentina
- Global Brain Health Institute, University of California, San Francisco, CA, USA; and Trinity College Dublin, Dublin, Ireland
| | - Cecilia González Campo
- National Scientific and Technical Research Council (CONICET), CABA, Buenos Aires, Argentina
- Cognitive Neuroscience Center (CNC), Universidad de San Andrés, Buenos Aires, Argentina
| | - Sebastián Moguilner
- Global Brain Health Institute, University of California, San Francisco, CA, USA; and Trinity College Dublin, Dublin, Ireland
- Latin American Brain Health Institute (BrainLat), Universidad Adolfo Ibáñez, Santiago, Chile
| | - Agustín Ibañez
- National Scientific and Technical Research Council (CONICET), CABA, Buenos Aires, Argentina
- Cognitive Neuroscience Center (CNC), Universidad de San Andrés, Buenos Aires, Argentina
- Global Brain Health Institute, University of California, San Francisco, CA, USA; and Trinity College Dublin, Dublin, Ireland
- Latin American Brain Health Institute (BrainLat), Universidad Adolfo Ibáñez, Santiago, Chile
- Trinity College Institute of Neuroscience (TCIN), Trinity College Dublin, Dublin, Ireland
| | - Enzo Tagliazucchi
- Department of Physics, University of Buenos Aires, Buenos Aires, Argentina
- National Scientific and Technical Research Council (CONICET), CABA, Buenos Aires, Argentina
- Cognitive Neuroscience Center (CNC), Universidad de San Andrés, Buenos Aires, Argentina
- Latin American Brain Health Institute (BrainLat), Universidad Adolfo Ibáñez, Santiago, Chile
| | - B. T. Thomas Yeo
- Centre for Sleep and Cognition, Centre for Translational MR Research, Department of Electrical and Computer Engineering, N.1 Institute for Health and Institute for Digital Medicine, National University of Singapore, Singapore
| | - Morten L. Kringelbach
- Department of Psychiatry, University of Oxford, Oxford, United Kingdom
- Center for Music in the Brain, Department of Clinical Medicine, Aarhus University, Aarhus, Denmark
- Life and Health Sciences Research Institute (ICVS), School of Medicine, University of Minho, Braga, Portugal
- Centre for Eudaimonia and Human Flourishing, University of Oxford, Oxford, United Kingdom
| | - Gustavo Deco
- Center for Brain and Cognition, Computational Neuroscience Group, Universitat Pompeu Fabra, Barcelona, Spain
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Barcelona, Spain
- Institució Catalana de la Recerca i Estudis Avancats (ICREA), Barcelona, Spain
- Department of Neuropsychology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany
- School of Psychological Sciences, Monash University, Melbourne, Australia
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18
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Murugadoss PR, Ambalarajan V, Sivakumar V, Dhandapani PB, Baleanu D. Analysis of Dengue Transmission Dynamic Model by Stability and Hopf Bifurcation with Two-Time Delays. FRONT BIOSCI-LANDMRK 2023; 28:117. [PMID: 37395028 DOI: 10.31083/j.fbl2806117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2023] [Revised: 04/26/2023] [Accepted: 05/05/2023] [Indexed: 07/04/2023]
Abstract
BACKGROUND Mathematical models reflecting the epidemiological dynamics of dengue infection have been discovered dating back to 1970. The four serotypes (DENV-1 to DENV-4) that cause dengue fever are antigenically related but different viruses that are transmitted by mosquitoes. It is a significant global public health issue since 2.5 billion individuals are at risk of contracting the virus. METHODS The purpose of this study is to carefully examine the transmission of dengue with a time delay. A dengue transmission dynamic model with two delays, the standard incidence, loss of immunity, recovery from infectiousness, and partial protection of the human population was developed. RESULTS Both endemic equilibrium and illness-free equilibrium were examined in terms of the stability theory of delay differential equations. As long as the basic reproduction number (R0) is less than unity, the illness-free equilibrium is locally asymptotically stable; however, when R0 exceeds unity, the equilibrium becomes unstable. The existence of Hopf bifurcation with delay as a bifurcation parameter and the conditions for endemic equilibrium stability were examined. To validate the theoretical results, numerical simulations were done. CONCLUSIONS The length of the time delay in the dengue transmission epidemic model has no effect on the stability of the illness-free equilibrium. Regardless, Hopf bifurcation may occur depending on how much the delay impacts the stability of the underlying equilibrium. This mathematical modelling is effective for providing qualitative evaluations for the recovery of a huge population of afflicted community members with a time delay.
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Affiliation(s)
- Prakash Raj Murugadoss
- Department of Mathematics, A.V.V.M. Sri Pushpam College (Affiliated to Bharathidasan University, Tiruchirappalli), Poondi, 613 503 Thanjavur, Tamil Nadu, India
| | - Venkatesh Ambalarajan
- Department of Mathematics, A.V.V.M. Sri Pushpam College (Affiliated to Bharathidasan University, Tiruchirappalli), Poondi, 613 503 Thanjavur, Tamil Nadu, India
| | - Vinoth Sivakumar
- Department of Mathematics, V.S.B. Engineering College, 639111 Karur, Tamil Nadu, India
| | | | - Dumitru Baleanu
- Department of Mathematics, Cankara University, 06530 Ankara, Turkey
- Institute of Space Sciences, Laboratory of Theoretical Physics, R 76900, Magurele-Bucharest, Romania
- Department of Natural Sciences, School of Arts and Sciences, Lebanese American University, 11022801 Beirut, Lebanon
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19
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Baccili AP, Monteiro LHA. Social Pressure from a Core Group can Cause Self-Sustained Oscillations in an Epidemic Model. Acta Biotheor 2023; 71:18. [PMID: 37347302 DOI: 10.1007/s10441-023-09469-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/12/2022] [Accepted: 06/08/2023] [Indexed: 06/23/2023]
Abstract
Let the individuals of a population be divided into two groups with different personal habits. The core group is associated with health risk behaviors; the non-core group avoids unhealthy activities. Assume that the infected individuals of the core group can spread a contagious disease to the whole population. Also, assume that cure does not confer immunity. Here, an epidemiological model written as a set of ordinary differential equations is proposed to investigate the infection propagation in this population. In the model, migrations between these two groups are allowed; however, the transitions from the non-core group into the core group prevail. These migrations can be either spontaneous or stimulated by social pressure. It is analytically shown that, in the scenario of spontaneous migration, the disease is either naturally eradicated or chronically persists at a constant level. In the scenario of stimulated migration, in addition to eradication and constant persistence, self-sustained oscillations in the number of sick individuals can also be found. These analytical results are illustrated by numerical simulations and discussed from a public health perspective.
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Affiliation(s)
- A P Baccili
- Universidade Presbiteriana Mackenzie, PPGEEC, Escola de Engenharia, Rua da Consolação, n.896, 01302-907, São Paulo, SP, Brazil
| | - L H A Monteiro
- Universidade Presbiteriana Mackenzie, PPGEEC, Escola de Engenharia, Rua da Consolação, n.896, 01302-907, São Paulo, SP, Brazil.
- Universidade de São Paulo, Escola Politécnica, São Paulo, SP, Brazil.
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20
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Meng Y, Namachchivaya NS, Perkowski N. Hopf Bifurcations of Moore-Greitzer PDE Model with Additive Noise. J Nonlinear Sci 2023; 33:74. [PMID: 37337607 PMCID: PMC10276801 DOI: 10.1007/s00332-023-09929-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/15/2021] [Accepted: 05/22/2023] [Indexed: 06/21/2023]
Abstract
The Moore-Greitzer partial differential equation (PDE) is a commonly used mathematical model for capturing flow and pressure changes in axial-flow jet engine compressors. Determined by compressor geometry, the deterministic model is characterized by three types of Hopf bifurcations as the throttle coefficient decreases, namely surge (mean flow oscillations), stall (inlet flow disturbances) or a combination of both. Instabilities place fundamental limits on jet-engine operating range and thus limit the design space. In contrast to the deterministic PDEs, the Hopf bifurcation in stochastic PDEs is not well understood. The goal of this particular work is to rigorously develop low-dimensional approximations using a multiscale analysis approach near the deterministic stall bifurcation points in the presence of additive noise acting on the fast modes. We also show that the reduced-dimensional approximations (SDEs) contain multiplicative noise. Instability margins in the presence of uncertainties can be thus approximated, which will eventually lead to lighter and more efficient jet engine design.
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Affiliation(s)
- Yiming Meng
- Department of Applied Mathematics, Waterloo University, Waterloo, ON Canada
| | | | - Nicolas Perkowski
- Institut für Mathematik, Freie Universität Berlin, Berlin, DE Germany
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21
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Zitouni NEH, Dellal M, Lakrib M. Substrate inhibition can produce coexistence and limit cycles in the chemostat model with allelopathy. J Math Biol 2023; 87:7. [PMID: 37311983 DOI: 10.1007/s00285-023-01943-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/29/2022] [Revised: 04/05/2023] [Accepted: 05/22/2023] [Indexed: 06/15/2023]
Abstract
In this work, we consider a model of two microbial species in a chemostat in which one of the competitors can produce a toxin (allelopathic agent) against the other competitor, and is itself inhibited by the substrate. The existence and stability conditions of all steady states of the reduced model in the plane are determined according to the operating parameters. With Michaelis-Menten or Monod growth functions, it is well known that the model can have a unique positive equilibrium which is unstable as long as it exists. By including both monotone and non-monotone growth functions (which is the case when there is substrate inhibition), it is shown that a new positive equilibrium point exists which can be stable according to the operating parameters of the system. This general model exhibits a rich behavior with the coexistence of two microbial species, the multi-stability, the occurrence of stable limit cycles through super-critical Hopf bifurcations and the saddle-node bifurcation of limit cycles. Moreover, the operating diagram describes some asymptotic behavior of this model by varying the operating parameters and illustrates the effect of the inhibition on the emergence of the coexistence region of the species.
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Affiliation(s)
| | - Mohamed Dellal
- Ibn Khaldoun University, 14000, Tiaret, Algeria.
- LDM, Djillali Liabès University, 22000, Sidi Bel Abbès, Algeria.
| | - Mustapha Lakrib
- LDM, Djillali Liabès University, 22000, Sidi Bel Abbès, Algeria
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22
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He J, Xiao M, Zhao J, Wang Z, Yao Y, Cao J. Tree-structured neural networks: Spatiotemporal dynamics and optimal control. Neural Netw 2023; 164:395-407. [PMID: 37172459 DOI: 10.1016/j.neunet.2023.04.039] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/27/2022] [Revised: 03/29/2023] [Accepted: 04/20/2023] [Indexed: 05/15/2023]
Abstract
How the network topology drives the response dynamic is a basic question that has not yet been fully answered in neural networks. Elucidating the internal relation between topological structures and dynamics is instrumental in our understanding of brain function. Recent studies have revealed that the ring structure and star structure have a great influence on the dynamical behavior of neural networks. In order to further explore the role of topological structures in the response dynamic, we construct a new tree structure that differs from the ring structure and star structure of traditional neural networks. Considering the diffusion effect, we propose a diffusion neural network model with binary tree structure and multiple delays. How to design control strategies to optimize brain function has also been an open question. Thus, we put forward a novel full-dimensional nonlinear state feedback control strategy to optimize relevant neurodynamics. Some conditions about the local stability and Hopf bifurcation are obtained, and it is proved that the Turing instability does not occur. Moreover, for the formation of the spatially homogeneous periodic solution, some diffusion conditions are also fused together. Finally, several numerical examples are carried out to illustrate the results' correctness. Meanwhile, some comparative experiments are rendered to reveal the effectiveness of the proposed control strategy.
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Affiliation(s)
- Jiajin He
- College of Automation & College of Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing 210023, Jiangsu, China.
| | - Min Xiao
- College of Automation & College of Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing 210023, Jiangsu, China.
| | - Jing Zhao
- College of Automation & College of Artificial Intelligence, Nanjing University of Posts and Telecommunications, Nanjing 210023, Jiangsu, China.
| | - Zhengxin Wang
- School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China.
| | - Yi Yao
- School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China.
| | - Jinde Cao
- School of Mathematics, Southeast University, Nanjing 210096, China; Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea.
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23
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Xue L, Zhang H, Zheng X, Sun W, Lei J. Treatment of melanoma with dendritic cell vaccines and immune checkpoint inhibitors: A mathematical modeling study. J Theor Biol 2023; 568:111489. [PMID: 37054970 DOI: 10.1016/j.jtbi.2023.111489] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/01/2022] [Revised: 12/13/2022] [Accepted: 04/05/2023] [Indexed: 04/15/2023]
Abstract
Dendritic cell (DC) vaccines and immune checkpoint inhibitors (ICIs) play critical roles in shaping the immune responses of tumor cells (TCs) and are widely used in cancer immunotherapies. Quantitatively evaluating the effectiveness of these therapies are essential for the optimization of treatment strategies. Here, based on the combined therapy of melanoma with DC vaccines and ICIs, we formulated a mathematical model to investigate the dynamic interactions between TCs and the immune system and understand the underlying mechanisms of immunotherapy. First, we obtained a threshold parameter for the growth of TCs, which is given by the ratio of spontaneous proliferation to immune inhibition. Next, we proved the existence and locally asymptotic stability of steady states of tumor-free, tumor-dominant, and tumor-immune coexistent equilibrium, and identified the existence of Hopf bifurcation of the proposed model. Furthermore, global sensitivity analysis showed that the growth of TCs strongly correlates with the injection rate of DC vaccines, the activation rate of CTLs, and the killing rate of TCs. Finally, we tested the efficacy of multiple monotherapies and combined therapies with model simulations. Our results indicate that DC vaccines can decelerate the growth of TCs, and ICIs can inhibit the growth of TCs. Besides, both therapies can prolong the lifetime of patients, and the combined therapy of DC vaccines and ICIs can effectively eradicate TCs.
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Affiliation(s)
- Ling Xue
- College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin, Heilongjiang, 150001, China; College of Mathematical Sciences, Harbin Engineering University, Harbin, Heilongjiang, 150001, China
| | - Hongyu Zhang
- College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin, Heilongjiang, 150001, China; College of Mathematical Sciences, Harbin Engineering University, Harbin, Heilongjiang, 150001, China
| | - Xiaoming Zheng
- Department of Mathematics, Central Michigan University, Mount Pleasant, MI 48859, United States of America
| | - Wei Sun
- College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin, Heilongjiang, 150001, China; College of Mathematical Sciences, Harbin Engineering University, Harbin, Heilongjiang, 150001, China.
| | - Jinzhi Lei
- School of Mathematical Sciences, Center for Applied Mathematics, Tiangong University, Tianjin, 300387, China.
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24
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Lu Y, Xin X, Rinzel J. Bistability at the onset of neuronal oscillations. Biol Cybern 2023; 117:61-79. [PMID: 36622415 DOI: 10.1007/s00422-022-00954-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/11/2022] [Accepted: 12/08/2022] [Indexed: 05/05/2023]
Abstract
The Hodgkin-Huxley (HH) model and squid axon (bathed in reduced Ca2+) fire repetitively for steady current injection. Moreover, for a current-range just suprathreshold, repetitive firing coexists with a stable steady state. Neuronal excitability, as such, shows bistability and hysteresis providing the opportunity for the system to perform as switchable between firing and non-firing states with transient input and providing the backbone as a dynamical mechanism for bursting oscillations. Some conditions for bistability can be derived by intricate analysis (bifurcation theory) and characterized by simulation, but conditions for emergence and robustness of such bistability do not typically follow from intuition. Here, we demonstrate with a semi-quantitative two-variable, V-w, reduction of the HH model features that promote/reduce bistability. Visualization of flow and trajectories in the V-w phase plane provides an intuitive grasp for bistability. The geometry of action potential recovery involves a late phase during which the dynamic negative feedback of [Formula: see text] inactivation and [Formula: see text] activation over/undershoot, respectively, their resting values, thereby leading to hyperexcitabilty and an intrinsically generated opportunity to by-pass the spiral-like stable rest state and initiate the next spike upstroke. We illustrate control of bistability and dependence of the degree of hysteresis on recovery timescales and gating properties. Our dynamical dissection reveals the strongly attracting depolarized phase of the spike, enabling approximations like the resetting feature of adapting integrate-and-fire models. We extend our insights and show that the Morris-Lecar model can also exhibit robust bistability.
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Affiliation(s)
- Yiqing Lu
- Courant Institute of Mathematical Sciences, New York University, New York, NY, USA
| | - Xiu Xin
- Shizuishan City, Ningxia, China
| | - John Rinzel
- Courant Institute of Mathematical Sciences, New York University, New York, NY, USA.
- Center for Neural Science, New York University, 4 Washington Place, New York, NY, 10003, USA.
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25
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Lu M, Gao D, Huang J, Wang H. Relative prevalence-based dispersal in an epidemic patch model. J Math Biol 2023; 86:52. [PMID: 36877332 PMCID: PMC9987411 DOI: 10.1007/s00285-023-01887-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2022] [Revised: 01/21/2023] [Accepted: 02/11/2023] [Indexed: 03/07/2023]
Abstract
In this paper, we propose a two-patch SIRS model with a nonlinear incidence rate: [Formula: see text] and nonconstant dispersal rates, where the dispersal rates of susceptible and recovered individuals depend on the relative disease prevalence in two patches. In an isolated environment, the model admits Bogdanov-Takens bifurcation of codimension 3 (cusp case) and Hopf bifurcation of codimension up to 2 as the parameters vary, and exhibits rich dynamics such as multiple coexistent steady states and periodic orbits, homoclinic orbits and multitype bistability. The long-term dynamics can be classified in terms of the infection rates [Formula: see text] (due to single contact) and [Formula: see text] (due to double exposures). In a connected environment, we establish a threshold [Formula: see text] between disease extinction and uniform persistence under certain conditions. We numerically explore the effect of population dispersal on disease spread when [Formula: see text] and patch 1 has a lower infection rate, our results indicate: (i) [Formula: see text] can be nonmonotonic in dispersal rates and [Formula: see text] ([Formula: see text] is the basic reproduction number of patch i) may fail; (ii) the constant dispersal of susceptible individuals (or infective individuals) between two patches (or from patch 2 to patch 1) will increase (or reduce) the overall disease prevalence; (iii) the relative prevalence-based dispersal may reduce the overall disease prevalence. When [Formula: see text] and the disease outbreaks periodically in each isolated patch, we find that: (a) small unidirectional and constant dispersal can lead to complex periodic patterns like relaxation oscillations or mixed-mode oscillations, whereas large ones can make the disease go extinct in one patch and persist in the form of a positive steady state or a periodic solution in the other patch; (b) relative prevalence-based and unidirectional dispersal can make periodic outbreak earlier.
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Affiliation(s)
- Min Lu
- School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079 Hubei People’s Republic of China
| | - Daozhou Gao
- Present Address: Department of Mathematics, Cleveland State University, Cleveland, 44115 Ohio USA
- Department of Mathematics, Shanghai Normal University, Shanghai, 200234 People’s Republic of China
| | - Jicai Huang
- School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079 Hubei People’s Republic of China
| | - Hao Wang
- Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1 AB Canada
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26
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Iwasa Y, Hayashi R. Waves of infection emerging from coupled social and epidemiological dynamics. J Theor Biol 2023; 558:111366. [PMID: 36435215 PMCID: PMC9682870 DOI: 10.1016/j.jtbi.2022.111366] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2022] [Revised: 11/16/2022] [Accepted: 11/18/2022] [Indexed: 11/24/2022]
Abstract
The coronavirus (SARS-CoV-2) exhibited waves of infection in 2020 and 2021 in Japan. The number of infected had multiple distinct peaks at intervals of several months. One possible process causing these waves of infection is people switching their activities in response to the prevalence of infection. In this paper, we present a simple model for the coupling of social and epidemiological dynamics. The assumptions are as follows. Each person switches between active and restrained states. Active people move more often to crowded areas, interact with each other, and suffer a higher rate of infection than people in the restrained state. The rate of transition from restrained to active states is enhanced by the fraction of currently active people (conformity), whereas the rate of backward transition is enhanced by the abundance of infected people (risk avoidance). The model may show transient or sustained oscillations, initial-condition dependence, and various bifurcations. The infection is maintained at a low level if the recovery rate is between the maximum and minimum levels of the force of infection. In addition, waves of infection may emerge instead of converging to the stationary abundance of infected people if both conformity and risk avoidance of people are strong.
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Affiliation(s)
- Yoh Iwasa
- Department of Biology, Faculty of Science, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan; Institute of Freshwater Biology, Nagano University, 1088 Komaki, Ueda, Agano 386-0031, Japan.
| | - Rena Hayashi
- Department of Biology, Faculty of Science, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan
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Wang J, Shen J. Turing instability mechanism of short-memory formation in multilayer FitzHugh-Nagumo network. Front Psychiatry 2023; 14:1083015. [PMID: 37051165 PMCID: PMC10083418 DOI: 10.3389/fpsyt.2023.1083015] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 10/28/2022] [Accepted: 02/14/2023] [Indexed: 04/14/2023] Open
Abstract
Introduction The study of brain function has been favored by scientists, but the mechanism of short-term memory formation has yet to be precise. Research problem Since the formation of short-term memories depends on neuronal activity, we try to explain the mechanism from the neuron level in this paper. Research contents and methods Due to the modular structures of the brain, we analyze the pattern properties of the FitzHugh-Nagumo model (FHN) on a multilayer network (coupled by a random network). The conditions of short-term memory formation in the multilayer FHN model are obtained. Then the time delay is introduced to more closely match patterns of brain activity. The properties of periodic solutions are obtained by the central manifold theorem. Conclusion When the diffusion coeffcient, noise intensity np, and network connection probability p reach a specific range, the brain forms a relatively vague memory. It is found that network and time delay can induce complex cluster dynamics. And the synchrony increases with the increase of p. That is, short-term memory becomes clearer.
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Affiliation(s)
- Junjie Wang
- School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, China
| | - Jianwei Shen
- School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou, China
- *Correspondence: Jianwei Shen
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28
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Sun Y, Chen Y, Zhang H, Chai Y. Dynamic effect of electromagnetic induction on epileptic waveform. BMC Neurosci 2022; 23:78. [PMID: 36536272 PMCID: PMC9764561 DOI: 10.1186/s12868-022-00768-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2022] [Accepted: 12/12/2022] [Indexed: 12/24/2022] Open
Abstract
BACKGROUND Electromagnetic induction has recently been considered as an important factor affecting the activity of neurons. However, as an important form of intervention in epilepsy treatment, few people have linked the two, especially the related dynamic mechanisms have not been explained clearly. METHODS Considering that electromagnetic induction has some brain area dependence, we proposed a modified two-compartment cortical thalamus model and set eight different key bifurcation parameters to study the transition mechanisms of epilepsy. We compared and analyzed the application and getting rid of memristors of single-compartment and coupled models. In particular, we plotted bifurcation diagrams to analyze the dynamic mechanisms behind abundant discharge activities, which mainly involved Hopf bifurcations (HB), fold of cycle bifurcations (LPC) and torus bifurcations (TR). RESULTS The results show that the coupled model can trigger more discharge states due to the driving effect between compartments. Moreover, the most remarkable finding of this study is that the memristor shows two sides. On the one hand, it may reduce tonic discharges. On the other hand, it may cause new pathological states. CONCLUSIONS The work explains the control effect of memristors on different brain regions and lays a theoretical foundation for future targeted therapy. Finally, it is hoped that our findings will provide new insights into the role of electromagnetic induction in absence seizures.
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Affiliation(s)
- Yuqin Sun
- grid.440635.00000 0000 9527 0839School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 201306 China
| | - Yuting Chen
- grid.440635.00000 0000 9527 0839School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 201306 China
| | - Hudong Zhang
- grid.440635.00000 0000 9527 0839School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 201306 China
| | - Yuan Chai
- grid.440635.00000 0000 9527 0839School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 201306 China
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29
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Zhang S, Li F, Xu X. Dynamics and control strategy for a delayed viral infection model. J Biol Dyn 2022; 16:44-63. [PMID: 35075973 DOI: 10.1080/17513758.2022.2028024] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/29/2020] [Accepted: 01/04/2022] [Indexed: 06/14/2023]
Abstract
In this paper, we derive a delayed epidemic model to describe the characterization of cytotoxic T lymphocyte (CTL)-mediated immune response against virus infection. The stability of equilibria and the existence of Hopf bifurcation are analysed. Theoretical results reveal that if the basic reproductive number is greater than 1, the positive equilibrium may lose its stability and the bifurcated periodic solution occurs when time delay is taken as the bifurcation parameter. Furthermore, we investigate an optimal control problem according to the delayed model based on the available therapy for hepatitis B infection. With the aim of minimizing the infected cells and viral load with consideration for the treatment costs, the optimal solution is discussed analytically. For the case when periodic solution occurs, numerical simulations are performed to suggest the optimal therapeutic strategy and compare the model-predicted consequences.
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Affiliation(s)
- Suxia Zhang
- School of Science, Xi'an University of Technology, Xi'an, People's Republic of China
| | - Fei Li
- School of Science, Xi'an University of Technology, Xi'an, People's Republic of China
| | - Xiaxia Xu
- School of Science, Xi'an University of Technology, Xi'an, People's Republic of China
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30
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Opoku-Sarkodie R, Bartha FA, Polner M, Röst G. Dynamics of an SIRWS model with waning of immunity and varying immune boosting period. J Biol Dyn 2022; 16:596-618. [PMID: 35943129 DOI: 10.1080/17513758.2022.2109766] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/18/2022] [Accepted: 07/29/2022] [Indexed: 06/15/2023]
Abstract
SIRS models capture transmission dynamics of infectious diseases for which immunity is not lifelong. Extending these models by a W compartment for individuals with waning immunity, the boosting of the immune system upon repeated exposure may be incorporated. Previous analyses assumed identical waning rates from R to W and from W to S. This implicitly assumes equal length for the period of full immunity and of waned immunity. We relax this restriction, and allow an asymmetric partitioning of the total immune period. Stability switches of the endemic equilibrium are investigated with a combination of analytic and numerical tools. Then, continuation methods are applied to track bifurcations along the equilibrium branch. We find rich dynamics: Hopf bifurcations, endemic double bubbles, and regions of bistability. Our results highlight that the length of the period in which waning immunity can be boosted is a crucial parameter significantly influencing long term epidemiological dynamics.
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Affiliation(s)
| | | | - Mónika Polner
- Bolyai Institute, University of Szeged, Szeged, Hungary
| | - Gergely Röst
- Bolyai Institute, University of Szeged, Szeged, Hungary
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31
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Chen M, Zheng Q, Wu R, Chen L. Hopf bifurcation in delayed nutrient-microorganism model with network structure. J Biol Dyn 2022; 16:1-13. [PMID: 35000575 DOI: 10.1080/17513758.2021.2020915] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2021] [Accepted: 11/23/2021] [Indexed: 06/14/2023]
Abstract
In this paper, we introduce and deal with the delayed nutrient-microorganism model with a random network structure. By employing time delay τ as the main critical value of the Hopf bifurcation, we investigate the direction of the Hopf bifurcation of such a random network nutrient-microorganism model. Noticing that the results of the direction of the Hopf bifurcation in a random network model are rare, we thus try to use the method of multiple time scales (MTS) to derive amplitude equation and determine the direction of the Hopf bifurcation. It is showed that the delayed random network nutrient-microorganism model can exhibit a supercritical or subcritical Hopf bifurcation. Numerical experiments are performed to verify the validity of the theoretical analysis.
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Affiliation(s)
- Mengxin Chen
- College of Mathematics and Information Science, Henan Normal University, Xinxiang, People's Republic of China
| | - Qianqian Zheng
- School of Mathematics and Statistics, Xuchang University, Xuchang, People's Republic of China
| | - Ranchao Wu
- School of Mathematical Sciences, Anhui University, Hefei, People's Republic of China
| | - Liping Chen
- School of Electrical Engineering and Automation, Hefei University of Technology, Hefei, People's Republic of China
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32
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Liao M, Liu Y, Liu S, Meyad AM. Stability and Hopf bifurcation of HIV-1 model with Holling II infection rate and immune delay. J Biol Dyn 2022; 16:397-411. [PMID: 33683178 DOI: 10.1080/17513758.2021.1895334] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/2020] [Accepted: 02/16/2021] [Indexed: 06/12/2023]
Abstract
This paper aims to analyse stability and Hopf bifurcation of the HIV-1 model with immune delay under the functional response of the Holling II type. The global stability analysis has been considered by Lyapunov-LaSalle theorem. And stability and the sufficient condition for the existence of Hopf Bifurcation of the infected equilibrium of the HIV-1 model with immune response are also studied. Some numerical simulations verify the above results. Finally, we propose a novel three dimension system to the future study.
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Affiliation(s)
- Maoxin Liao
- School of Mathematics and Physics, University of South China, Hengyang, People's Republic of China
| | - Yanjin Liu
- School of Mathematics and Physics, University of South China, Hengyang, People's Republic of China
| | - Shinan Liu
- School of Mathematics and Physics, University of South China, Hengyang, People's Republic of China
| | - Ali M Meyad
- School of Public Finance and Taxation, Southwestern University of Finance and Economics, Chengdu, People's Republic of China
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33
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Liu J, Guo Z, Guo H. The blood-stage dynamics of malaria infection with immune response. J Biol Dyn 2022; 16:294-319. [PMID: 34936537 DOI: 10.1080/17513758.2021.2017033] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2021] [Accepted: 11/30/2021] [Indexed: 06/14/2023]
Abstract
This article is concerned with the dynamics of malaria infection model with diffusion and delay. The disease free threshold ℜ0 and the immune response threshold value ℜ1 of the malaria infection are obtained, which characterize the stability of the disease free equilibrium and infection equilibrium (with or without immune response). In addition, fluctuations occur when the model undergoes Hopf bifurcation as the delay passes through a certain critical value τ0. In this case, periodic oscillation appears near the positive steady state, which implies the recurrent attacks of disease. Finally, numerical simulations are provided to illustrate the theoretical results.
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Affiliation(s)
- Jian Liu
- Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China
- School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, People's Republic of China
| | - Zhiming Guo
- Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China
- School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, People's Republic of China
| | - Hongpeng Guo
- Center for Applied Mathematics, Guangzhou University, Guangzhou, People's Republic of China
- School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, People's Republic of China
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34
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Abstract
In this paper, we are concerned with an epidemic model with quarantine and distributed time delay. We define the basic reproduction number R0 and show that if R0≤1, then the disease-free equilibrium is globally asymptotically stable, whereas if R0>1, then it is unstable and there exists a unique endemic equilibrium. We obtain sufficient conditions for a Hopf bifurcation that induces a nontrivial periodic solution which represents recurrent epidemic waves. By numerical simulations, we illustrate stability and instability parameter regions. Our results suggest that the quarantine and time delay play important roles in the occurrence of recurrent epidemic waves.
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Affiliation(s)
- Toshikazu Kuniya
- Graduate School of System Informatics, Kobe University, Kobe, Japan
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35
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He M, Li Z. Stability of a fear effect predator-prey model with mutual interference or group defense. J Biol Dyn 2022; 16:480-498. [PMID: 35759246 DOI: 10.1080/17513758.2022.2091800] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/2021] [Accepted: 06/03/2022] [Indexed: 06/15/2023]
Abstract
In this paper, we consider a fear effect predator-prey model with mutual interference or group defense. For the model with mutual interference, we show the interior equilibrium is globally stable, and the mutual interference can stabilize the predator-prey system. For the model with group defense, we discuss the singular dynamics around the origin and the occurrence of Hopf bifurcation, and find that there is a separatrix curve near the origin such that the orbits above which tend to the origin and the orbits below which tend to limit cycle or the interior equilibrium.
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Affiliation(s)
- Mengxin He
- College of Mathematics and Data Science, Minjiang University, Fuzhou, People's Republic of China
| | - Zhong Li
- School of Mathematics and Statistics, Fuzhou University, Fuzhou, People's Republic of China
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36
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Sun D, Liu J, Su X, Pei G. Fractional differential equation modeling of the HBV infection with time delay and logistic proliferation. Front Public Health 2022; 10:1036901. [PMID: 36438294 PMCID: PMC9692119 DOI: 10.3389/fpubh.2022.1036901] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2022] [Accepted: 09/27/2022] [Indexed: 11/13/2022] Open
Abstract
In this article, a fractional-order differential equation model of HBV infection was proposed with a Caputo derivative, delayed immune response, and logistic proliferation. Initially, infection-free and infection equilibriums and the basic reproduction number were computed. Thereafter, the stability of the two equilibriums was analyzed based on the fractional Routh-Hurwitz stability criterion, and the results indicated that the stability will change if the time delay or fractional order changes. In addition, the sensitivity of the basic reproduction number was analyzed to find out the most sensitive parameter. Lastly, the theoretical analysis was verified by numerical simulations. The results showed that the time delay of immune response and fractional order can significantly affect the dynamic behavior in the HBV infection process. Therefore, it is necessary to consider time delay and fractional order in modeling HBV infection and studying its dynamics.
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Affiliation(s)
- Deshun Sun
- Intelligent Medical Innovation Center, Southern University of Science and Technology Hospital, Shenzhen, China,Shenzhen Key Laboratory of Tissue Engineering, Shenzhen Second People's Hospital (The First Hospital Affiliated to Shenzhen University, Health Science Center), Shenzhen, China,*Correspondence: Deshun Sun
| | - Jingxiang Liu
- School of Marine Electrical Engineering, Dalian Maritime University, Dalian, China
| | - Xiuyun Su
- Intelligent Medical Innovation Center, Southern University of Science and Technology Hospital, Shenzhen, China
| | - Guoxian Pei
- Intelligent Medical Innovation Center, Southern University of Science and Technology Hospital, Shenzhen, China,Guoxian Pei
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37
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Zhang H, Wei J. Bifurcation analysis for a single population model with advection. J Math Biol 2022; 85:61. [PMID: 36305980 DOI: 10.1007/s00285-022-01818-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2021] [Revised: 06/25/2022] [Accepted: 09/26/2022] [Indexed: 12/29/2022]
Abstract
In this paper, the dynamics of a single population model with a general growth function is investigated in an advective environment. We show the existence of a nonconstant positive steady state, and give sufficient conditions for the occurrence of a Hopf bifurcation at the positive steady state. Moreover, the theoretical results are applied to the diffusive Nicholson's blowflies and Mackey-Glass's models with advection and delay, respectively. We numerically show that the population density decreases as the increase of advection rate or death rate, and a delay-induced Hopf bifurcation is more likely to occur with small advection or low mortality rate.
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Affiliation(s)
- Hua Zhang
- Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong, 264209, People's Republic of China
| | - Junjie Wei
- Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong, 264209, People's Republic of China. .,School of Science, Jimei University, Xiamen, Fujian, 361021, People's Republic of China.
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38
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Ghosh U, Thirthar AA, Mondal B, Majumdar P. Effect of Fear, Treatment, and Hunting Cooperation on an Eco-Epidemiological Model: Memory Effect in Terms of Fractional Derivative. Iran J Sci Technol Trans A Sci 2022; 46:1541-1554. [PMID: 36320931 PMCID: PMC9607782 DOI: 10.1007/s40995-022-01371-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/14/2022] [Accepted: 09/30/2022] [Indexed: 11/06/2022]
Abstract
In this paper, we have studied a fractional-order eco-epidemiological model incorporating fear, treatment, and hunting cooperation effects to explore the memory effect in the ecological system through Caputo-type fractional-order derivative. We have studied the behavior of different equilibrium points with the memory effect. The proposed system undergoes through Hopf bifurcation with respect to the memory parameter as the bifurcation parameter. We perform numerical simulations for different values of the memory parameter and some of model parameters. In the numerical results, it appears that the system is exhibiting a stable behavior from a period or chaotic nature with the increase in the memory effect. The system also exhibits two transcritical bifurcations with respect to the growth rate of the prey. At low values of prey's growth, all species go to extinction, at moderate values of prey's growth, only preys (susceptible and infected) can survive, and at higher values of prey's growth, all species survive simultaneously. The paper ended with some recommendations.
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Affiliation(s)
- Uttam Ghosh
- Department of Applied Mathematics, University of Calcutta, Kolkata, 700009 India
| | | | - Bapin Mondal
- Department of Applied Mathematics, University of Calcutta, Kolkata, 700009 India
| | - Prahlad Majumdar
- Department of Applied Mathematics, University of Calcutta, Kolkata, 700009 India
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39
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Li C, Ma Z, Wang Y. Dynamics of a delayed rumor spreading model with discontinuous threshold control. Heliyon 2022; 8:e11231. [PMID: 36311360 PMCID: PMC9615325 DOI: 10.1016/j.heliyon.2022.e11231] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/26/2022] [Revised: 05/26/2022] [Accepted: 10/19/2022] [Indexed: 11/06/2022] Open
Abstract
In this paper, we studied a delayed rumor spreading model with discontinuous threshold control. First, we studied the existence of equilibria of the subsystem. Regarding the delay as bifurcating parameter, the local asymptotic stability and Hopf bifurcation of the positive equilibrium are discussed by analyzing the corresponding characteristic equations of linearized systems. Then, we studied the existence of the sliding mode and analyzed the existence of the tangent equilibria, boundary equilibria, regular equilibria, and the stability of the pseudo-equilibrium. Finally, we provide some numerical simulations to verify the theoretical results.
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Affiliation(s)
- Chunru Li
- Business School, Changshu Institute of Technology, Changshu, 215500, China,Corresponding author.
| | - Zujun Ma
- School of Economics and Management, Southwest Jiaotong University, Chengdu, 610031, China
| | - Yuanyuan Wang
- Business School, Changshu Institute of Technology, Changshu, 215500, China
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40
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Nouaoura S, Fekih-Salem R, Abdellatif N, Sari T. Operating diagrams for a three-tiered microbial food web in the chemostat. J Math Biol 2022; 85:44. [PMID: 36190574 DOI: 10.1007/s00285-022-01812-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2021] [Revised: 06/07/2022] [Accepted: 06/10/2022] [Indexed: 10/10/2022]
Abstract
In this paper, we consider a three-tiered food web model in a chemostat, including chlorophenol, phenol, and hydrogen substrates and their degraders. The model takes into account the three substrate inflowing concentrations, as well as maintenance, that is, decay terms of the species. The operating diagrams give the asymptotic behavior of the model with respect to the four operating parameters, which are the dilution rate and the three inflowing concentrations of the substrates. These diagrams were obtained only numerically in the existing literature. Using the mathematical analysis of this model obtained in our previous studies, we construct the operating diagrams, by plotting the curves that separate their various regions. Hence, the regions of the operating diagrams are constructed analytically and there is no requirement for time-consuming algorithms to generate the plots, as in the numerical method. Moreover, our method reveals behaviors that have not been detected in the previous numerical studies. The growth functions are of Monod form with the inclusion of a product inhibition term. However, our method applies for a large class of growth functions. We construct operating diagrams with and without maintenance showing the role of maintenance on the stability of the system.
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41
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Zhao L, Huang C, Cao J. Effects of double delays on bifurcation for a fractional-order neural network. Cogn Neurodyn 2022; 16:1189-1201. [PMID: 36237412 PMCID: PMC9508308 DOI: 10.1007/s11571-021-09762-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/06/2021] [Revised: 10/25/2021] [Accepted: 11/22/2021] [Indexed: 11/03/2022] Open
Abstract
Neural network bifurcation is an important nonlinear dynamic behavior of neural network, which plays an important role in cognitive calculation. The effects of leakage delay or communication delay on the stability and bifurcation of a fractional-order neural network (FONN) are researched. By viewing leakage delay or communication delay as the bifurcation parameters to detect the bifurcations conditions of the developed FONN, respectively, we capture the bifurcation points with regard to leakage delay or communication delay. It alleges that FONN exhibits excellent stability performance with choosing smaller values of them, and Hopf bifurcations emerge of FONN and induce poor performance if selecting a larger ones. In the end, numerical examples are employed to evaluate the feasibleness of the analytical discoveries.
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Affiliation(s)
- Lingzhi Zhao
- School of Information Engineering, Nanjing Xiaozhuang University, Nanjing, 211171 China
| | - Chengdai Huang
- School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000 China
| | - Jinde Cao
- School of Mathematics, Southeast University, Nanjing, 210096 China
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42
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Xu C, Liu Z, Aouiti C, Li P, Yao L, Yan J. New exploration on bifurcation for fractional-order quaternion-valued neural networks involving leakage delays. Cogn Neurodyn 2022; 16:1233-1248. [PMID: 36237401 PMCID: PMC9508321 DOI: 10.1007/s11571-021-09763-1] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2021] [Revised: 10/16/2021] [Accepted: 11/27/2021] [Indexed: 11/24/2022] Open
Abstract
During the past decades, many works on Hopf bifurcation of fractional-order neural networks are mainly concerned with real-valued and complex-valued cases. However, few publications involve the quaternion-valued neural networks which is a generalization of real-valued and complex-valued neural networks. In this present study, we explorate the Hopf bifurcation problem for fractional-order quaternion-valued neural networks involving leakage delays. Taking advantage of the Hamilton rule of quaternion algebra, we decompose the addressed fractional-order quaternion-valued delayed neural networks into the equivalent eight real valued networks. Then the delay-inspired bifurcation condition of the eight real valued networks are derived by making use of the stability criterion and bifurcation theory of fractional-order differential dynamical systems. The impact of leakage delay on the bifurcation behavior of the involved fractional-order quaternion-valued delayed neural networks has been revealed. Software simulations are implemented to support the effectiveness of the derived fruits of this study. The research supplements the work of Huang et al. (Neural Netw 117:67-93, 2019).
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Affiliation(s)
- Changjin Xu
- Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang, 550004 People’s Republic of China
- Guizhou Key Laboratory of Big Data Statistical Analysis, Guiyang, 550025 People’s Republic of China
| | - Zixin Liu
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550004 People’s Republic of China
| | - Chaouki Aouiti
- Faculty of Sciences of Bizerta, UR13ES47 Research Units of Mathematics and Applications, University of Carthage, Bizerta, 7021 Tunisia
| | - Peiluan Li
- School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023 People’s Republic of China
| | - Lingyun Yao
- Library, Guizhou University of Finance and Economics, Guiyang, 550004 People’s Republic of China
| | - Jinling Yan
- School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023 People’s Republic of China
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43
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Xiong L, Garfinkel A. A common pathway to cancer: Oncogenic mutations abolish p53 oscillations. Prog Biophys Mol Biol 2022; 174:28-40. [PMID: 35752348 DOI: 10.1016/j.pbiomolbio.2022.06.002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2021] [Revised: 06/13/2022] [Accepted: 06/20/2022] [Indexed: 06/15/2023]
Abstract
The tumor suppressor p53 oscillates in response to DNA double-strand breaks, a behavior that has been suggested to be essential to its anti-cancer function. Nearly all human cancers have genetic alterations in the p53 pathway; a number of these alterations have been shown to be oncogenic by experiment. These alterations include somatic mutations and copy number variations as well as germline polymorphisms. Intriguingly, they exhibit a mixed pattern of interactions in tumors, such as co-occurrence, mutual exclusivity, and paradoxically, mutual antagonism. Using a differential equation model of p53-Mdm2 dynamics, we employ Hopf bifurcation analysis to show that these alterations have a common mode of action, to abolish the oscillatory competence of p53, thereby, we suggest, impairing its tumor suppressive function. In this analysis, diverse genetic alterations, widely associated with human cancers clinically, have a unified mechanistic explanation of their role in oncogenesis.
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Affiliation(s)
- Lingyun Xiong
- Department of Stem Cell Biology and Regenerative Medicine, University of Southern California, Los Angeles, CA 90007 USA; Department of Quantitative and Computational Biology, University of Southern California, Los Angeles, CA, 90007, USA; Ludwig Institute for Cancer Research, University of Oxford, Oxford, OX3 7DQ, UK
| | - Alan Garfinkel
- Departments of Medicine (Cardiology) and Integrative Biology and Physiology, University of California, Los Angeles, CA, 90095, USA; Newton-Abraham Visiting Professor (2019-2020), Lincoln College and Department of Computer Science, University of Oxford, Oxford, OX1 3DR, UK.
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44
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Blair DS, Soriano-Mas C, Cabral J, Moreira P, Morgado P, Deco G. Complexity changes in functional state dynamics suggest focal connectivity reductions. Front Hum Neurosci 2022; 16:958706. [PMID: 36211126 PMCID: PMC9540393 DOI: 10.3389/fnhum.2022.958706] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2022] [Accepted: 08/03/2022] [Indexed: 11/13/2022] Open
Abstract
The past two decades have seen an explosion in the methods and directions of neuroscience research. Along with many others, complexity research has rapidly gained traction as both an independent research field and a valuable subdiscipline in computational neuroscience. In the past decade alone, several studies have suggested that psychiatric disorders affect the spatiotemporal complexity of both global and region-specific brain activity (Liu et al., 2013; Adhikari et al., 2017; Li et al., 2018). However, many of these studies have not accounted for the distributed nature of cognition in either the global or regional complexity estimates, which may lead to erroneous interpretations of both global and region-specific entropy estimates. To alleviate this concern, we propose a novel method for estimating complexity. This method relies upon projecting dynamic functional connectivity into a low-dimensional space which captures the distributed nature of brain activity. Dimension-specific entropy may be estimated within this space, which in turn allows for a rapid estimate of global signal complexity. Testing this method on a recently acquired obsessive-compulsive disorder dataset reveals substantial increases in the complexity of both global and dimension-specific activity versus healthy controls, suggesting that obsessive-compulsive patients may experience increased disorder in cognition. To probe the potential causes of this alteration, we estimate subject-level effective connectivity via a Hopf oscillator-based model dynamic model, the results of which suggest that obsessive-compulsive patients may experience abnormally high connectivity across a broad network in the cortex. These findings are broadly in line with results from previous studies, suggesting that this method is both robust and sensitive to group-level complexity alterations.
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Affiliation(s)
| | - Carles Soriano-Mas
- Psychiatry and Mental Health Group, Neuroscience Program, Institut d’Investigació Biomèdica de Bellvitge, Barcelona, Spain
- Network Center for Biomedical Research on Mental Health, Carlos III Health Institute, Madrid, Spain
- Department of Social Psychology and Quantitative Psychology, Universitat de Barcelona, Barcelona, Spain
| | - Joana Cabral
- Life and Health Sciences Research Institute, School of Medicine, University of Minho, Braga, Portugal
| | - Pedro Moreira
- Life and Health Sciences Research Institute, School of Medicine, University of Minho, Braga, Portugal
- ICVS/3B’s, PT Government Associate Laboratory, Braga, Portugal
- Psychological Neuroscience Lab, CIPsi, School of Psychology, University of Minho, Braga, Portugal
| | - Pedro Morgado
- Life and Health Sciences Research Institute, School of Medicine, University of Minho, Braga, Portugal
- ICVS/3B’s, PT Government Associate Laboratory, Braga, Portugal
- Clinical Academic Center—Braga, Braga, Portugal
| | - Gustavo Deco
- Facultad de Comunicación, Universitat Pompeu Fabra, Barcelona, Spain
- Institució Catalana de Recerca i Estudis Avançats, Barcelona, Spain
- Department of Neuropsychology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany
- School of Psychological Sciences, Monash University, Clayton, VIC, Australia
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45
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Song P, Xiao Y. Analysis of a diffusive epidemic system with spatial heterogeneity and lag effect of media impact. J Math Biol 2022; 85:17. [PMID: 35913603 PMCID: PMC9340761 DOI: 10.1007/s00285-022-01780-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2021] [Revised: 04/26/2022] [Accepted: 06/24/2022] [Indexed: 12/02/2022]
Abstract
We considered an SIS functional partial differential model cooperated with spatial heterogeneity and lag effect of media impact. The wellposedness including existence and uniqueness of the solution was proved. We defined the basic reproduction number and investigated the threshold dynamics of the model, and discussed the asymptotic behavior and monotonicity of the basic reproduction number associated with the diffusion rate. The local and global Hopf bifurcation at the endemic steady state was investigated theoretically and numerically. There exists numerical cases showing that the larger the number of basic reproduction number, the smaller the final epidemic size. The meaningful conclusion generalizes the previous conclusion of ordinary differential equation.
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Affiliation(s)
- Pengfei Song
- Department of Applied Mathematics, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, PR China
| | - Yanni Xiao
- Department of Applied Mathematics, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, PR China.
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46
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Ghakanyuy BM, Teboh-Ewungkem MI, Schneider KA, Ngwa GA. Investigating the impact of multiple feeding attempts on mosquito dynamics via mathematical models. Math Biosci 2022; 350:108832. [PMID: 35718220 DOI: 10.1016/j.mbs.2022.108832] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/08/2021] [Revised: 04/07/2022] [Accepted: 05/09/2022] [Indexed: 11/27/2022]
Abstract
A deterministic differential equation model for the dynamics of terrestrial forms of mosquito populations is studied. The model assesses the impact of multiple probing attempts by mosquitoes that quest for blood within human populations by including a waiting class for mosquitoes that failed a blood feeding attempt. The equations are derived based on the idea that the reproductive cycle of the mosquito can be viewed as a set of alternating egg laying and blood feeding outcomes realized on a directed path called the gonotrophic cycle pathway. There exists a threshold parameter, the basic offspring number for mosquitoes, whose nature is affected by the way we interpret the transitions involving the different classes on the gonotrophic cycle path. The trivial steady state for the system, which always exists, can be globally asymptomatically stable whenever the threshold parameter is less than unity. The non-trivial steady state, when it exists, is stable for a range of values of the threshold parameter but can also be driven to instability via a Hopf bifurcation. The model's output reveals that the waiting class mosquitoes do contribute positively to sustain mosquito populations as well as increase their interactions with humans via increased frequency and initial amplitude of oscillations. We conclude that to understand human-mosquito interactions, it is informative to consider multiple probing attempts; known to occur when mosquitoes quest for blood meals within human populations.
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Affiliation(s)
- Bime M Ghakanyuy
- Department of Mathematics, University of Buea, P.O. Box 63, Buea, Cameroon
| | | | - Kristan A Schneider
- Department of Applied Computer and Bio-Sciences, University of Applied Sciences, Mittweida, Technikumplatz 17, 09648 Mittweida, Germany
| | - Gideon A Ngwa
- Department of Mathematics, University of Buea, P.O. Box 63, Buea, Cameroon.
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47
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Chakraborty A. A remark on "COVID-19: Perturbation dynamics resulting chaos to stable with seasonality transmission" [Chaos, Solitons and Fractals 145 (2021) 110772]. Chaos Solitons Fractals 2022; 156:111831. [PMID: 35095220 PMCID: PMC8784539 DOI: 10.1016/j.chaos.2022.111831] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/09/2021] [Revised: 01/13/2022] [Accepted: 01/19/2022] [Indexed: 06/14/2023]
Abstract
In Batabyal (2021)[2], introducing an extension of the well-known susceptible-exposed-infected-recovered (SEIR) model with seasonality transmission of SARS-CoV-2, the author has derived and discussed various analytical and numerical results. Careful scrutiny of the said article brings about some genuine issues pertaining to the model formulation, analysis and numerical studies carried out in Batabyal (2021)[2]. Given the present pandemic and the havoc it has been causing throughout the world, and the responsibility of giving out rightful information/results backed by scientific proofs, there is a pressing need to address issues of such kind right away.
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Affiliation(s)
- Avishek Chakraborty
- Department of Engineering Science, Academy Of Technology, Adisaptagram, Krishnapur-Chandanpur, Hooghly-712502, India
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48
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Cui X, Xue D, Li T. Fractional-order delayed Ross-Macdonald model for malaria transmission. Nonlinear Dyn 2022; 107:3155-3173. [PMID: 35095196 PMCID: PMC8782717 DOI: 10.1007/s11071-021-07114-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/25/2021] [Accepted: 11/28/2021] [Indexed: 06/14/2023]
Abstract
This paper proposes a novel fractional-order delayed Ross-Macdonald model for malaria transmission. This paper aims to systematically investigate the effect of both the incubation periods of Plasmodium and the order on the dynamic behavior of diseases. Utilizing inequality techniques, contraction mapping theory, fractional linear stability theorem, and bifurcation theory, several sufficient conditions for the existence and uniqueness of solutions, the local stability of the positive equilibrium point, and the existence of fractional-order Hopf bifurcation are obtained under different time delays cases. The results show that time delay can change the stability of system. System becomes unstable and generates a Hopf bifurcation when the delay increases to a certain value. Besides, the value of order influences the stability interval size. Thus, incubation periods and the order have a major effect on the dynamic behavior of the model. The effectiveness of the theoretical results is shown through numerical simulations.
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Affiliation(s)
- Xinshu Cui
- College of Information Science and Engineering, Northeastern University, Shenyang, 110819 Liaoning China
| | - Dingyu Xue
- College of Information Science and Engineering, Northeastern University, Shenyang, 110819 Liaoning China
| | - Tingxue Li
- College of Information Science and Engineering, Northeastern University, Shenyang, 110819 Liaoning China
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Li T, Xiao Y. Complex dynamics of an epidemic model with saturated media coverage and recovery. Nonlinear Dyn 2022; 107:2995-3023. [PMID: 35068691 PMCID: PMC8761114 DOI: 10.1007/s11071-021-07096-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 06/21/2021] [Accepted: 11/21/2021] [Indexed: 06/14/2023]
Abstract
During the outbreak of emerging infectious diseases, media coverage and medical resource play important roles in affecting the disease transmission. To investigate the effects of the saturation of media coverage and limited medical resources, we proposed a mathematical model with extra compartment of media coverage and two nonlinear functions. We theoretically and numerically investigate the dynamics of the proposed model. Given great difficulties caused by high nonlinearity in theoretical analysis, we separately considered subsystems with only nonlinear recovery or with only saturated media impact. For the model with only nonlinear recovery, we theoretically showed that backward bifurcation can occur and multiple equilibria may coexist under certain conditions in this case. Numerical simulations reveal the rich dynamic behaviors, including forward-backward bifurcation, Hopf bifurcation, saddle-node bifurcation, homoclinic bifurcation and unstable limit cycle. So the limitation of medical resources induces rich dynamics and causes much difficulties in eliminating the infectious diseases. We then investigated the dynamics of the system with only saturated media impact and concluded that saturated media impact hardly induces the complicated dynamics. Further, we parameterized the proposed model on the basis of the COVID-19 case data in mainland China and data related to news items, and estimated the basic reproduction number to be 2.86. Sensitivity analyses were carried out to quantify the relative importance of parameters in determining the cumulative number of infected individuals at the end of the first month of the outbreak. Combining with numerical analyses, we suggested that providing adequate medical resources and improving media response to infection or individuals' response to mass media may reduce the cumulative number of the infected individuals, which mitigates the transmission dynamics during the early stage of the COVID-19 pandemic.
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Affiliation(s)
- Tangjuan Li
- School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049 Shaanxi China
| | - Yanni Xiao
- School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049 Shaanxi China
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50
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Majee S, Adak S, Jana S, Mandal M, Kar TK. Complex dynamics of a fractional-order SIR system in the context of COVID-19. J Appl Math Comput 2022; 68:4051-4074. [PMID: 35043050 PMCID: PMC8758247 DOI: 10.1007/s12190-021-01681-z] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/24/2021] [Revised: 11/27/2021] [Accepted: 11/30/2021] [Indexed: 06/14/2023]
Abstract
This paper proposes and analyses a new fractional-order SIR type epidemic model with a saturated treatment function. The detailed dynamics of the corresponding system, including the equilibrium points and their existence and uniqueness, uniform-boundedness, and stability of the solutions are studied. The threshold parameter, basic reproduction number of the system which determines the disease dynamics is derived, and the condition of occurrence of backward bifurcation is also determined. Some numerical works are conducted to validate our analytical results for the commensurate fractional-order system. Hopf bifurcations for the fractional-order system are studied by taking the order of the fractional differential as a bifurcation parameter.
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Affiliation(s)
- Suvankar Majee
- Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal 711103 India
| | - Sayani Adak
- Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal 711103 India
| | - Soovoojeet Jana
- Department of Mathematics, Ramsaday College, Amta, Howrah, West Bengal 711401 India
| | - Manotosh Mandal
- Department of Mathematics, Tamralipta Mahavidyalaya, Tamluk, West Bengal 721636 India
| | - T. K. Kar
- Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal 711103 India
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