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Fujiwara N, Onaga T, Wada T, Takeuchi S, Seto J, Nakaya T, Aihara K. Analytical estimation of maximum fraction of infected individuals with one-shot non-pharmaceutical intervention in a hybrid epidemic model. BMC Infect Dis 2022; 22:512. [PMID: 35650534 PMCID: PMC9157046 DOI: 10.1186/s12879-022-07403-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/02/2021] [Accepted: 04/19/2022] [Indexed: 11/13/2022] Open
Abstract
BACKGROUND Facing a global epidemic of new infectious diseases such as COVID-19, non-pharmaceutical interventions (NPIs), which reduce transmission rates without medical actions, are being implemented around the world to mitigate spreads. One of the problems in assessing the effects of NPIs is that different NPIs have been implemented at different times based on the situation of each country; therefore, few assumptions can be shared about how the introduction of policies affects the patient population. Mathematical models can contribute to further understanding these phenomena by obtaining analytical solutions as well as numerical simulations. METHODS AND RESULTS In this study, an NPI was introduced into the SIR model for a conceptual study of infectious diseases under the condition that the transmission rate was reduced to a fixed value only once within a finite time duration, and its effect was analyzed numerically and theoretically. It was analytically shown that the maximum fraction of infected individuals and the final size could be larger if the intervention starts too early. The analytical results also suggested that more individuals may be infected at the peak of the second wave with a stronger intervention. CONCLUSIONS This study provides quantitative relationship between the strength of a one-shot intervention and the reduction in the number of patients with no approximation. This suggests the importance of the strength and time of NPIs, although detailed studies are necessary for the implementation of NPIs in complicated real-world environments as the model used in this study is based on various simplifications.
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Affiliation(s)
- Naoya Fujiwara
- Graduate School of Information Sciences, Tohoku University, 6-3-09 Aoba, Aramaki-aza Aoba-ku, Sendai, 980-8579, Miyagi, Japan.
- PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho, Kawaguchi, 332-0012, Saitama, Japan.
- Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, 153-8505, Tokyo, Japan.
- Center for Spatial Information Science, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8508, Chiba, Japan.
| | - Tomokatsu Onaga
- Graduate School of Information Sciences, Tohoku University, 6-3-09 Aoba, Aramaki-aza Aoba-ku, Sendai, 980-8579, Miyagi, Japan
- Frontier Research Institute for Interdisciplinary Sciences, Tohoku University, Aramaki aza Aoba 6-3, Aoba-ku, Sendai, 980-8578, Miyagi, Japan
| | - Takayuki Wada
- Department of Microbiology, Graduate School of Human Life and Ecology, Osaka Metropolitan University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Osaka, Japan
| | - Shouhei Takeuchi
- Faculty of Nursing and Nutrition, University of Nagasaki, 1-1-1 Manabino, Nagayo-cho, Nishi-Sonogi-gun, Nagasaki, 851-2195, Japan
| | - Junji Seto
- Department of Microbiology, Yamagata Prefectural Institute of Public Health, 1-6-6 Toka-machi, Yamagata, 990-0031, Yamagata, Japan
| | - Tomoki Nakaya
- Graduate School of Environmental Studies, Tohoku University, Aoba, 468-1, Aramaki, Aoba-ku, Sendai, 980-8572, Miyagi, Japan
| | - Kazuyuki Aihara
- International Research Center for Neurointelligence, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-0033, Tokyo, Japan
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Dong C, Xiang C, Qin W, Yang Y. Global dynamics for a Filippov system with media effects. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:2835-2852. [PMID: 35240809 DOI: 10.3934/mbe.2022130] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
In the process of spreading infectious diseases, the media accelerates the dissemination of information, and people have a deeper understanding of the disease, which will significantly change their behavior and reduce the disease transmission; it is very beneficial for people to prevent and control diseases effectively. We propose a Filippov epidemic model with nonlinear incidence to describe media's influence in the epidemic transmission process. Our proposed model extends existing models by introducing a threshold strategy to describe the effects of media coverage once the number of infected individuals exceeds a threshold. Meanwhile, we perform the stability of the equilibriua, boundary equilibrium bifurcation, and global dynamics. The system shows complex dynamical behaviors and eventually stabilizes at the equilibrium points of the subsystem or pseudo equilibrium. In addition, numerical simulation results show that choosing appropriate thresholds and control intensity can stop infectious disease outbreaks, and media coverage can reduce the burden of disease outbreaks and shorten the duration of disease eruptions.
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Affiliation(s)
- Cunjuan Dong
- School of Mathematics and Statistics, Hubei Minzu University, Enshi, Hubei 445000, China
| | - Changcheng Xiang
- School of Mathematics and Statistics, Hubei Minzu University, Enshi, Hubei 445000, China
| | - Wenjin Qin
- School of Mathematics and Statistics, Hubei Minzu University, Enshi, Hubei 445000, China
| | - Yi Yang
- College of Computer Science and Engineering, Chongqing Three Gorges University, Chongqing 404020, China
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Chladná Z, Kopfová J, Rachinskii D, Rouf SC. Global dynamics of SIR model with switched transmission rate. J Math Biol 2020; 80:1209-1233. [PMID: 31900507 DOI: 10.1007/s00285-019-01460-2] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/22/2019] [Revised: 10/14/2019] [Indexed: 10/25/2022]
Abstract
We propose a new epidemiological model, based on the classical SIR model, taking additionally into account a switching prevention strategy. The model has two distinct thresholds that determine the beginning and the end of an intervention and two different transmission rates. We study the global dynamics of the proposed two-dimensional model.
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Affiliation(s)
- Zuzana Chladná
- Department of Applied Mathematics and Statistics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 84248, Bratislava, Slovakia
| | - Jana Kopfová
- Mathematical Institute of the Silesian University, NaRybníčku 1, 746 01, Opava, Czech Republic
| | - Dmitrii Rachinskii
- University of Texas at Dallas, 800 W. Campbell, Richardson, TX, 75080, USA
| | - Samiha C Rouf
- University of Texas at Dallas, 800 W. Campbell, Richardson, TX, 75080, USA.
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Qin W, Tan X, Shi X, Xiang C. IPM strategies to a discrete switching predator-prey model induced by a mate-finding Allee effect. JOURNAL OF BIOLOGICAL DYNAMICS 2019; 13:586-605. [PMID: 31686604 DOI: 10.1080/17513758.2019.1682200] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2018] [Accepted: 08/24/2019] [Indexed: 06/10/2023]
Abstract
This paper proposes a discrete switching predator-prey model with a mate-finding Allee effect, where also switches are guided by Allee effect. One of the strategies analysed is to use a chemical in order to prevent the pest outbreak when the pest population is free of Allee effect. In this paper, we first study analytically the dynamic behaviors of the two subsystems and the equilibria and their stability of the switched system. Then we provide numerical bifurcation analyses for the switched discrete system. These show that the switched discrete system may have very complex dynamics by 2-parameter bifurcation diagrams which divide the space into regions and study equilibria, and 1-dimensional bifurcation diagrams which reveal that the system has periodic, chaotic solutions, period doubling bifurcations and so on. Furthermore, we try to refer the key parameters and initial densities of both populations associated with pest outbreaks and study their biological implications.
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Affiliation(s)
- Wenjie Qin
- Three Gorges Mathematics Research Center, China Three Gorges University, Yichang, People's Republic of China
- Engineering Research Center of Eco-environment in Three Gorges Reservoir Region of Ministry of Education, China Three Gorges University, Yichang, People's Republic of China
| | - Xuewen Tan
- School of Mathematics and Computer Science, Yunnan Minzu University, Kunming, People's Republic of China
| | - Xiaotao Shi
- Engineering Research Center of Eco-environment in Three Gorges Reservoir Region of Ministry of Education, China Three Gorges University, Yichang, People's Republic of China
| | - Changcheng Xiang
- College of Science, Hubei Minzu University, Enshi, People's Republic of China
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Wang A, Xiao Y, Smith R. Multiple Equilibria in a Non-smooth Epidemic Model with Medical-Resource Constraints. Bull Math Biol 2018; 81:963-994. [PMID: 30539326 DOI: 10.1007/s11538-018-00544-2] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/27/2017] [Accepted: 11/28/2018] [Indexed: 12/01/2022]
Abstract
The issue of medical-resource constraints has the potential to dramatically affect disease management, especially in developing countries. We analyze a non-smooth epidemic model with nonlinear incidence rate and resource constraints, which defines a vaccination program with vaccination rate proportional to the number of susceptible individuals when this number is below the threshold level and constant otherwise. To better understand the impact of this non-smooth vaccination policy, we provide a comprehensive qualitative analysis of global dynamics for the whole parameter space. As the threshold value varies, the target model admits multistability of three regular equilibria, bistability of two regular equilibria, that of one disease-free equilibrium and one generalized endemic equilibria, and that of one disease-free equilibrium and one crossing cycle. The steady-state regimes include healthy, low epidemic and high epidemic. This suggests the key role of the threshold value, as well as the initial infection condition in disease control. Our findings demonstrate that the case number can be contained at a satisfactorily controllable level or range if eradicating it proves to be impossible.
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Affiliation(s)
- Aili Wang
- School of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, 721013, People's Republic of China
| | - Yanni Xiao
- School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, People's Republic of China
| | - Robert Smith
- Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada.
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Qin W, Tang S, Xiang C, Yang Y. Effects of limited medical resource on a Filippov infectious disease model induced by selection pressure. APPLIED MATHEMATICS AND COMPUTATION 2016; 283:339-354. [PMID: 32287500 PMCID: PMC7126627 DOI: 10.1016/j.amc.2016.02.042] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/11/2015] [Revised: 11/26/2015] [Accepted: 02/21/2016] [Indexed: 06/11/2023]
Abstract
In reality, the outbreak of emerging infectious diseases including SARS, A/H1N1 and Ebola are accompanied by the common cold and flu. The selective treatment measure for mitigating and controlling the emerging infectious diseases should be implemented due to limited medical resources. However, how to determine the threshold infected cases and when to implement the selective treatment tactics are crucial for disease control. To address this, we derive a non-smooth Filippov system induced by selective treatment measure. The dynamic behaviors of two subsystems have been discussed completely, and the existence conditions for sliding segment, sliding mode dynamics and different types of equilibria such as regular equilibrium, pseudo-equilibrium, boundary equilibrium and tangent point have been provided. Further, numerical sliding bifurcation analyses show that the proposed Filippov system has rich sliding bifurcations. Especially, the most interesting results are those for the fixed parameter set as the bifurcation parameter varies, the sliding bifurcations occur sequentially: crossing → buckling → real/virtual equilibrium → buckling → crossing. The key factors which affect the selective treatment measures and the threshold value of infected cases for emerging infectious disease have been discussed in more detail.
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Affiliation(s)
- Wenjie Qin
- College of Science, China Three Gorges University, Yichang 443002, PR China
| | - Sanyi Tang
- College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, PR China
| | - Changcheng Xiang
- Department of Mathematics, Hubei University for Nationalities, Enshi 445000, PR China
| | - Yali Yang
- College of Science, Air Force Engineering University, Xi’an 710051, PR China
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