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Haghighi A. A Truncated Split-Step Forward Euler–Maruyama-Based Method for Stochastic Systems with Non-globally Lipschitz Coefficients. IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY, TRANSACTIONS A: SCIENCE 2022. [DOI: 10.1007/s40995-022-01399-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
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2
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Zarin R. Modeling and numerical analysis of fractional order hepatitis B virus model with harmonic mean type incidence rate. Comput Methods Biomech Biomed Engin 2022:1-16. [DOI: 10.1080/10255842.2022.2103371] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/07/2023]
Affiliation(s)
- Rahat Zarin
- Department of Basic Sciences, University of Engineering and Technology Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan
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Liu X, Arfan M, Ur Rahman M, Fatima B. Analysis of SIQR type mathematical model under Atangana-Baleanu fractional differential operator. Comput Methods Biomech Biomed Engin 2022; 26:98-112. [PMID: 35271386 DOI: 10.1080/10255842.2022.2047954] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
Abstract
In the given manuscript, the fractional mathematical model for the current pandemic of COVID-19 is investigated. The model is composed of four agents of susceptible (S), infectious (I), quarantined (Q) and recovered (R) cases respectively. The fractional operator of Atangana-Baleanu-Caputo (ABC) is applied to the considered model for the fractional dynamics. The basic reproduction number is computed for the stability analysis. The techniques of existence and uniqueness of the solution are established with the help of fixed point theory. The concept of stability is also derived using the Ulam-Hyers stability technique. With the help of the fractional order numerical method of Adams-Bashforth, we find the approximate solution of the said model. The obtained scheme is simulated on different fractional orders along with the comparison of integer orders. Varying the numerical values for the contact rate ζ, different simulations are performed to check the effect of it on the dynamics of COVID-19.
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Affiliation(s)
- Xuan Liu
- Department of Mathematics, Hanshan Normal University, Chaozhou, China
| | - Muhammad Arfan
- Department of Mathematics, Government Degree College Gulabad Dir (L), Khyber Pakhtunkhwa, Pakistan
| | - Mati Ur Rahman
- School of Mathematical Science, Shanghai Jiao Tong University, Shanghai, P.R. China
| | - Bibi Fatima
- Department of Mathematics, University of Malakand, Chakdara Dir (L), Khyber Pakhtunkhwa, Pakistan
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Nawaz Y, Arif MS, Ashraf MU. Development of Explicit Schemes for Diffusive SEAIR COVID-19 Epidemic Spreading Model: An Application to Computational Biology. IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY. TRANSACTION A, SCIENCE 2021; 45:2109-2119. [PMID: 34539151 PMCID: PMC8435406 DOI: 10.1007/s40995-021-01214-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 06/18/2021] [Accepted: 08/25/2021] [Indexed: 10/28/2022]
Abstract
In this contribution, a first-order time scheme is proposed for finding solutions to partial differential equations (PDEs). A mathematical model of the COVID-19 epidemic is modified where the recovery rate of exposed individuals is also considered. The linear stability of the equilibrium states for the modified COVID-19 model is given by finding its Jacobian and applying Routh-Hurwitz criteria on characteristic polynomial. The proposed scheme provides the first-order accuracy in time and second-order accuracy in space. The stability of the proposed scheme is given using the von Neumann stability criterion for standard parabolic PDEs. The consistency for the proposed scheme is also given by expanding the involved terms in it using the Taylor series. The scheme can be used to obtain the condition of getting a positive solution. The stability region of the scheme can be enlarged by choosing suitable values of the contained parameter. Finally, a comparison of the proposed scheme is made with the existing non-standard finite difference method. The results indicate that the non-standard classical technique is incapable of preserving the unique characteristics of the model's epidemiologically significant solutions, whereas the proposed approaches are capable of doing so. A computational code for the proposed discrete model scheme may be made available to readers upon request for convenience.
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Affiliation(s)
- Yasir Nawaz
- Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000 Pakistan
| | - Muhammad Shoaib Arif
- Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000 Pakistan
| | - Muhammad Usman Ashraf
- Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Islamabad, Pakistan
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Kang T, Zhang Q. Dynamics of a stochastic delayed avian influenza model with mutation and temporary immunity. INT J BIOMATH 2021. [DOI: 10.1142/s1793524521500297] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, the dynamic behaviors are studied for a stochastic delayed avian influenza model with mutation and temporary immunity. First, we prove the existence and uniqueness of the global positive solution for the stochastic model. Second, we give two different thresholds [Formula: see text] and [Formula: see text], and further establish the sufficient conditions of extinction and persistence in the mean for the avian-only subsystem and avian-human system, respectively. Compared with the corresponding deterministic model, the thresholds affected by the white noises are smaller than the ones of the deterministic system. Finally, numerical simulations are carried out to support our theoretical results. It is concluded that the vaccination immunity period can suppress the spread of avian influenza during poultry and human populations, while prompt the spread of mutant avian influenza in human population.
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Affiliation(s)
- Ting Kang
- School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, P. R. China
- Xinhua College, Ningxia University, Yinchuan 750021, P. R. China
| | - Qimin Zhang
- School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, P. R. China
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Raza A, Ahmadian A, Rafiq M, Salahshour S, Ferrara M. An analysis of a nonlinear susceptible-exposed-infected-quarantine-recovered pandemic model of a novel coronavirus with delay effect. RESULTS IN PHYSICS 2021; 21:103771. [PMID: 33391985 PMCID: PMC7768216 DOI: 10.1016/j.rinp.2020.103771] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/18/2020] [Revised: 12/21/2020] [Accepted: 12/21/2020] [Indexed: 05/03/2023]
Abstract
In the present study, a nonlinear delayed coronavirus pandemic model is investigated in the human population. For study, we find the equilibria of susceptible-exposed-infected-quarantine-recovered model with delay term. The stability of the model is investigated using well-posedness, Routh Hurwitz criterion, Volterra Lyapunov function, and Lasalle invariance principle. The effect of the reproduction number on dynamics of disease is analyzed. If the reproduction number is less than one then the disease has been controlled. On the other hand, if the reproduction number is greater than one then the disease has become endemic in the population. The effect of the quarantine component on the reproduction number is also investigated. In the delayed analysis of the model, we investigated that transmission dynamics of the disease is dependent on delay terms which is also reflected in basic reproduction number. At the end, to depict the strength of the theoretical analysis of the model, computer simulations are presented.
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Affiliation(s)
- Ali Raza
- Department of Mathematics, National College of Business Administration and Economics Lahore, Pakistan
| | - Ali Ahmadian
- Institute of Visual Informatics, National University of Malaysia, 43600 UKM, Bangi, Selangor, Malaysia
- Department of Law, Economics and Human Sciences& Decisions Lab, University Mediterranea of Reggio Calabria, 89125 Reggio Calabria, Italy
- School of Mathematical Sciences, College of Science and Technology, Wenzhou-Kean University, Wenzhou, China
| | - Muhammad Rafiq
- Department of Mathematics, Faculty of Sciences, University of Central Punjab, Lahore, Pakistan
| | - Soheil Salahshour
- Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey
| | - Massimiliano Ferrara
- Department of Law, Economics and Human Sciences& Decisions Lab, University Mediterranea of Reggio Calabria, 89125 Reggio Calabria, Italy
- ICRIOS-The Invernizzi Centre for Research in Innovation, Organization, Strategy and Entrepreneurship, Bocconi University, Department of Management and Technology Via Sarfatti, 25 20136 Milano MI, Italy
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Das S, Ghosh P, Banerjee S, Pyne S, Chattopadhyay J, Mukhopadhyay I. Determination of critical community size from an HIV/AIDS model. PLoS One 2021; 16:e0244543. [PMID: 33507898 PMCID: PMC7842972 DOI: 10.1371/journal.pone.0244543] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/03/2020] [Accepted: 12/11/2020] [Indexed: 11/19/2022] Open
Abstract
After an epidemic outbreak, the infection persists in a community long enough to engulf the entire susceptible population. Local extinction of the disease could be possible if the susceptible population gets depleted. In large communities, the tendency of eventual damp down of recurrent epidemics is balanced by random variability. But, in small communities, the infection would die out when the number of susceptible falls below a certain threshold. Critical community size (CCS) is considered to be the mentioned threshold, at which the infection is as likely as not to die out after a major epidemic for small communities unless reintroduced from outside. The determination of CCS could aid in devising systematic control strategies to eradicate the infectious disease from small communities. In this article, we have come up with a simplified computation based approach to deduce the CCS of HIV disease dynamics. We consider a deterministic HIV model proposed by Silva and Torres, and following Nåsell, introduce stochasticity in the model through time-varying population sizes of different compartments. Besides, Metcalf’s group observed that the relative risk of extinction of some infections on islands is almost double that in the mainlands i.e. infections cease to exist at a significantly higher rate in islands compared to the mainlands. They attributed this phenomenon to the greater recolonization in the mainlands. Interestingly, the application of our method on demographic facts and figures of countries in the AIDS belt of Africa led us to expect that existing control measures and isolated locations would assist in temporary eradication of HIV infection much faster. For example, our method suggests that through systematic control strategies, after 7.36 years HIV epidemics will temporarily be eradicated from different communes of island nation Madagascar, where the population size falls below its CCS value, unless the disease is reintroduced from outside.
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Affiliation(s)
- Sarmistha Das
- Human Genetics Unit, Indian Statistical Institute, Kolkata, West Bengal, India
| | - Pramit Ghosh
- Deben Mahata Government Medical College & Hospital, Purulia, West Bengal, India
| | - Sandip Banerjee
- Department of Mathematics, Indian Institute of Technology, Roorkee, Uttarakhand, India
| | - Saumyadipta Pyne
- Public Health Dynamics Lab and Department of Biostatistics, Graduate School of Public Health, University of Pittsburgh, Pittsburgh, Pennsylvania, United States of America
- Health Analytics Network, Pittsburgh, Pennsylvania, United States of America
| | - Joydev Chattopadhyay
- Agricultural and Ecological Research Unit, Indian Statistical Institute, Kolkata, West Bengal, India
| | - Indranil Mukhopadhyay
- Human Genetics Unit, Indian Statistical Institute, Kolkata, West Bengal, India
- * E-mail:
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Deshpande A, Samanta S, Govindarajan S, Layek RK. Multi-bit Boolean model for chemotactic drift of Escherichia coli. IET Syst Biol 2021; 14:343-349. [PMID: 33399098 DOI: 10.1049/iet-syb.2020.0060] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022] Open
Abstract
Dynamic biological systems can be modelled to an equivalent modular structure using Boolean networks (BNs) due to their simple construction and relative ease of integration. The chemotaxis network of the bacterium Escherichia coli (E. coli) is one of the most investigated biological systems. In this study, the authors developed a multi-bit Boolean approach to model the drifting behaviour of the E. coli chemotaxis system. Their approach, which is slightly different than the conventional BNs, is designed to provide finer resolution to mimic high-level functional behaviour. Using this approach, they simulated the transient and steady-state responses of the chemoreceptor sensory module. Furthermore, they estimated the drift velocity under conditions of the exponential nutrient gradient. Their predictions on chemotactic drifting are in good agreement with the experimental measurements under similar input conditions. Taken together, by simulating chemotactic drifting, they propose that multi-bit Boolean methodology can be used for modelling complex biological networks. Application of the method towards designing bio-inspired systems such as nano-bots is discussed.
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Affiliation(s)
- Anuj Deshpande
- Department of Electronics and Communication Engineering, SRM University - AP, Andhra Pradesh, India.
| | - Sibendu Samanta
- Department of Electronics and Communication Engineering, SRM University - AP, Andhra Pradesh, India
| | | | - Ritwik Kumar Layek
- Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology - Kharagpur, West Bengal, India
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Xin L, Guo Y, Zhu Q. Stochastic epidemic dynamics based on the association between susceptible and recovered individuals. INT J BIOMATH 2020. [DOI: 10.1142/s1793524520500850] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this paper, we propose a new mathematical model based on the association between susceptible and recovered individual. Then, we study the stability of this model with the deterministic case and obtain the conditions for the extinction of diseases. Moreover, in view of the association between susceptible and recovered individual perturbed by white noise, we also give sufficient conditions for the extinction and the permanence in mean of disease with the white noise. Finally, we have numerical simulations to demonstrate the correctness of obtained theoretical results.
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Affiliation(s)
- Luyao Xin
- School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, P. R. China
| | - Yingxin Guo
- School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, P. R. China
| | - Quanxin Zhu
- MOE-LCSM, School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, Hunan, P. R. China
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Raza A, Rafiq M, Baleanu D, Arif MS. Numerical simulations for stochastic meme epidemic model. ADVANCES IN DIFFERENCE EQUATIONS 2020. [PMID: 0 DOI: 10.1186/s13662-020-02593-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/04/2023]
Abstract
AbstractThe primary purpose of this study is to perform the comparison of deterministic and stochastic modeling. The effect of threshold number is also observed in this model. For numerical simulations, we have developed some stochastic explicit approaches, but they are dependent on time step size. The implicitly driven explicit approach has been developed for a stochastic meme model. The proposed approach is always independent of time step size. Also, we have presented theorems in support of convergence of the proposed approach for the stochastic meme model.
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11
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Wanduku D. The stationary distribution and stochastic persistence for a class of disease models: Case study of malaria. INT J BIOMATH 2020. [DOI: 10.1142/s1793524520500242] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
This paper presents a nonlinear family of stochastic SEIRS models for diseases such as malaria in a highly random environment with noises from the disease transmission and natural death rates, and also from the random delays of the incubation and immunity periods. Improved analytical methods and local martingale characterizations are applied to find conditions for the disease to persist near an endemic steady state, and also for the disease to remain permanently in the system over time. Moreover, the ergodic stationary distribution for the stochastic process describing the disease dynamics is defined, and the statistical characteristics of the distribution are given numerically. The results of this study show that the disease will persist and become permanent in the system, regardless of (1) whether the noises are from the disease transmission rate and/or from the natural death rates or (2) whether the delays in the system are constant or random for individuals in the system. Furthermore, it is shown that “weak” noise is associated with the existence of an endemic stationary distribution for the disease, while “strong” noise is associated with extinction of the population over time. Numerical simulation examples for Plasmodium vivax malaria are given.
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Affiliation(s)
- Divine Wanduku
- Department of Mathematical Sciences, Georgia Southern University, 65 Georgia Avenue, Room 3042, Statesboro, GA 30460, USA
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