1
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Sahani SK, Jakhad A. Incidence rate drive the multiple wave in the COVID-19 pandemic. MethodsX 2023; 11:102317. [PMID: 37637293 PMCID: PMC10457448 DOI: 10.1016/j.mex.2023.102317] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2023] [Accepted: 08/03/2023] [Indexed: 08/29/2023] Open
Abstract
The last three years have been the most challenging for humanity due to the COVID-19 pandemic. The novel viral infection has eventually been able to infect most of the human population. It is now considered to be in the endemic stage, meaning it will remain in our world throughout our lifetime. There will be an intermittent outbreak of the COVID infection from time to time. Therefore, it is necessary to formulate a robust Mathematical model to study the dynamics of disease to have a control mechanism in place. In this article, we suggest a modified MSEIR model to explain the dynamics of COVID-19 infection. We assume that a susceptible person contracting the coronavirus develops a transient immunity to the illness. Further, infectives comprise asymptomatic, symptomatic, hospitalized and quarantined individuals. We assume that the incidence rate is of standard type, and susceptible can only become infective if they come in contact with either asymptomatic or symptomatic individuals. This basic and simple model effectively models the various waves every country has seen during the Pandemic. The simple analysis shows that the model could suggest various waves in future if we carefully select the incidence rate for the infection. In summary, we have discussed the following major points in this article. •We have analysed for local behavior infection-free equilibrium solution. Further, a thorough numerical exploration with various parameter settings has been performed to obtain the different cases of infection dynamics of the coronavirus epidemic.•We have found some interesting scenarios which explain the emergence of multiple waves observed in many countries.
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Affiliation(s)
- Saroj Kumar Sahani
- Faculty of Mathematics and Computer Science, Department of Mathematics, South Asian University Akbar Bhawan, Chankyapuri, New Delhi, Delhi 110021, India
| | - Anjali Jakhad
- Faculty of Mathematics and Computer Science, Department of Mathematics, South Asian University Akbar Bhawan, Chankyapuri, New Delhi, Delhi 110021, India
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2
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Luo L, Li B, Wang X, Cui L, Liu G. Interpretable spatial identity neural network-based epidemic prediction. Sci Rep 2023; 13:18159. [PMID: 37875546 PMCID: PMC10598274 DOI: 10.1038/s41598-023-45177-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/30/2023] [Accepted: 10/17/2023] [Indexed: 10/26/2023] Open
Abstract
Epidemic spatial-temporal risk analysis, e.g., infectious number forecasting, is a mainstream task in the multivariate time series research field, which plays a crucial role in the public health management process. With the rise of deep learning methods, many studies have focused on the epidemic prediction problem. However, recent primary prediction techniques face two challenges: the overcomplicated model and unsatisfactory interpretability. Therefore, this paper proposes an Interpretable Spatial IDentity (ISID) neural network to predict infectious numbers at the regional weekly level, which employs a light model structure and provides post-hoc explanations. First, this paper streamlines the classical spatio-temporal identity model (STID) and retains the optional spatial identity matrix for learning the contagion relationship between regions. Second, the well-known SHapley Additive explanations (SHAP) method was adopted to interpret how the ISID model predicts with multivariate sliding-window time series input data. The prediction accuracy of ISID is compared with several models in the experimental study, and the results show that the proposed ISID model achieves satisfactory epidemic prediction performance. Furthermore, the SHAP result demonstrates that the ISID pays particular attention to the most proximate and remote data in the input sequence (typically 20 steps long) while paying little attention to the intermediate steps. This study contributes to reliable and interpretable epidemic prediction through a more coherent approach for public health experts.
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Affiliation(s)
- Lanjun Luo
- School of Management, North Sichuan Medical College, Nanchong, China
| | - Boxiao Li
- Graduate School of Information Sciences, Tohoku University, Sendai, Japan
| | - Xueyan Wang
- Information Centre, Affiliated Hospital of North Sichuan Medical College, Nanchong, China.
| | - Lei Cui
- School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, China
| | - Gang Liu
- School of Management, Huazhong University of Science and Technology, Wuhan, China
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3
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Bonnemain T, Butano M, Bonnet T, Echeverría-Huarte I, Seguin A, Nicolas A, Appert-Rolland C, Ullmo D. Pedestrians in static crowds are not grains, but game players. Phys Rev E 2023; 107:024612. [PMID: 36932629 DOI: 10.1103/physreve.107.024612] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/13/2022] [Accepted: 02/01/2023] [Indexed: 06/18/2023]
Abstract
The local navigation of pedestrians is assumed to involve no anticipation beyond the most imminent collisions, in most models. These typically fail to reproduce some key features experimentally evidenced in dense crowds crossed by an intruder, namely, transverse displacements toward regions of higher density due to the anticipation of the intruder's crossing. We introduce a minimal model based on mean-field games, emulating agents planning out a global strategy that minimizes their overall discomfort. By solving the problem in the permanent regime thanks to an elegant analogy with the nonlinear Schrödinger's equation, we are able to identify the two main variables governing the model's behavior and to exhaustively investigate its phase diagram. We find that, compared to some prominent microscopic approaches, the model is remarkably successful in replicating the experimental observations associated with the intruder experiment. In addition, the model can capture other daily-life situations such as partial metro boarding.
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Affiliation(s)
- Thibault Bonnemain
- Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, NE1 8ST, United Kingdom
- Department of Mathematics, King's College, London KCL WC2R 2LS, United Kingdom
| | - Matteo Butano
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
| | - Théophile Bonnet
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
- Université Paris-Saclay, CNRS, IJCLab, 91405 Orsay, France
| | - Iñaki Echeverría-Huarte
- Laboratorio de Medios Granulares, Departamento de Física y Matemática Aplicada, Univ. Navarra, 31080 Pamplona, Spain
| | - Antoine Seguin
- Université Paris-Saclay, CNRS, FAST, 91405 Orsay, France
| | - Alexandre Nicolas
- Institut Lumière Matière, CNRS & Université Claude Bernard Lyon 1, 69622 Villeurbanne, France
| | | | - Denis Ullmo
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
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4
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Yan Q, Cheke RA, Tang S. Coupling an individual adaptive-decision model with a SIRV model of influenza vaccination reveals new insights for epidemic control. Stat Med 2022; 42:716-729. [PMID: 36577149 PMCID: PMC9880662 DOI: 10.1002/sim.9639] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2022] [Revised: 11/08/2022] [Accepted: 12/16/2022] [Indexed: 12/29/2022]
Abstract
Past seasonal influenza epidemics and vaccination experience may affect individuals' decisions on whether to be vaccinated or not, decisions that may be constantly reassessed in relation to recent influenza related experience. To understand the potentially complex interaction between experience and decisions and whether the vaccination rate is likely to reach a critical coverage level or not, we construct an adaptive-decision model. This model is then coupled with an influenza vaccination dynamics (SIRV) model to explore the interaction between individuals' decision-making and an influenza epidemic. Nonlinear least squares estimation is used to obtain the best-fit parameter values in the SIRV model based on data on new influenza-like illness (ILI) cases in Texas. Uncertainty and sensitivity analyses are then carried out to determine the impact of key parameters of the adaptive decision-making model on the ILI epidemic. The results showed that the necessary critical coverage rate of ILI vaccination could not be reached by voluntary vaccination. However, it could be reached in the fourth year if mass media reports improved individuals' memory of past vaccination experience. Individuals' memory of past vaccination experience, the proportion with histories of past vaccinations and the perceived cost of vaccination are important factors determining whether an ILI epidemic can be effectively controlled or not. Therefore, health authorities should guide people to improve their memory of past vaccination experience through media reports, publish timely data on annual vaccination proportions and adjust relevant measures to appropriately reduce vaccination perceived cost, in order to effectively control an ILI epidemic.
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Affiliation(s)
- Qinling Yan
- School of ScienceChang'an UniversityXi'anPeople's Republic of China
| | - Robert A. Cheke
- Natural Resources InstituteUniversity of Greenwich at MedwayChatham MaritimeKentUK
| | - Sanyi Tang
- School of Mathematics and StatisticsShaanxi Normal UniversityXi'anPeople's Republic of China
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5
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Lee W, Liu S, Li W, Osher S. Mean field control problems for vaccine distribution. RESEARCH IN THE MATHEMATICAL SCIENCES 2022; 9:51. [PMID: 35915747 PMCID: PMC9328629 DOI: 10.1007/s40687-022-00350-2] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/22/2021] [Accepted: 07/07/2022] [Indexed: 06/15/2023]
Abstract
With the invention of the COVID-19 vaccine, shipping and distributing are crucial in controlling the pandemic. In this paper, we build a mean-field variational problem in a spatial domain, which controls the propagation of pandemics by the optimal transportation strategy of vaccine distribution. Here, we integrate the vaccine distribution into the mean-field SIR model designed in Lee W, Liu S, Tembine H, Li W, Osher S (2020) Controlling propagation of epidemics via mean-field games. arXiv preprint arXiv:2006.01249. Numerical examples demonstrate that the proposed model provides practical strategies for vaccine distribution in a spatial domain.
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Affiliation(s)
- Wonjun Lee
- Department of Mathematics, University of California, Los Angeles, USA
| | - Siting Liu
- Department of Mathematics, University of California, Los Angeles, USA
| | - Wuchen Li
- Department of Mathematics, University of South Carolina, Columbia, USA
| | - Stanley Osher
- Department of Mathematics, University of California, Los Angeles, USA
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6
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Oliva G, Schlueter M, Munetomo M, Scala A. Dynamical intervention planning against COVID-19-like epidemics. PLoS One 2022; 17:e0269830. [PMID: 35700170 PMCID: PMC9197046 DOI: 10.1371/journal.pone.0269830] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/13/2021] [Accepted: 05/30/2022] [Indexed: 11/28/2022] Open
Abstract
COVID-19 has got us to face a new situation where, for the lack of ready-to-use vaccines, it is necessary to support vaccination with complex non-pharmaceutical strategies. In this paper, we provide a novel Mixed Integer Nonlinear Programming formulation for fine-grained optimal intervention planning (i.e., at the level of the single day) against newborn epidemics like COVID-19, where a modified SIR model accounting for heterogeneous population classes, social distancing and several types of vaccines (each with its efficacy and delayed effects), allows us to plan an optimal mixed strategy (both pharmaceutical and non-pharmaceutical) that takes into account both the vaccine availability in limited batches at selected time instants and the need for second doses while keeping hospitalizations and intensive care occupancy below a threshold and requiring that new infections die out at the end of the planning horizon. In order to show the effectiveness of the proposed formulation, we analyze a case study for Italy with realistic parameters.
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Affiliation(s)
- Gabriele Oliva
- Unit of Automatic Control, Department of Engineering, Università Campus Bio-Medico di Roma, Rome, Italy
- * E-mail: (GO); (AS)
| | - Martin Schlueter
- Information Initiative Center, Hokkaido University, Sapporo, Japan
| | | | - Antonio Scala
- CNR-ISC, Applico Lab, Roma, Italy
- Centro Ricerche Enrico Fermi, Roma, Italy
- Big Data in Health Society, Roma, Italy
- Global Health Security Agenda - GHSA, Roma, Italy
- * E-mail: (GO); (AS)
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7
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Singh P, Gupta A. Generalized SIR (GSIR) epidemic model: An improved framework for the predictive monitoring of COVID-19 pandemic. ISA TRANSACTIONS 2022; 124:31-40. [PMID: 33610314 PMCID: PMC7883688 DOI: 10.1016/j.isatra.2021.02.016] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2020] [Revised: 12/30/2020] [Accepted: 02/11/2021] [Indexed: 05/08/2023]
Abstract
Novel coronavirus respiratory disease COVID-19 has caused havoc in many countries across the globe. In order to contain infection of this highly contagious disease, most of the world population is constrained to live in a complete or partial lockdown for months together with a minimal human-to-human interaction having far reaching consequences on countries' economy and mental well-being of their citizens. Hence, there is a need for a good predictive model for the health advisory bodies and decision makers for taking calculated proactive measures to contain the pandemic and maintain a healthy economy. This paper extends the mathematical theory of the classical Susceptible-Infected-Removed (SIR) epidemic model and proposes a Generalized SIR (GSIR) model that is an integrative model encompassing multiple waves of daily reported cases. Existing growth function models of epidemic have been shown as the special cases of the GSIR model. Dynamic modeling of the parameters reflect the impact of policy decisions, social awareness, and the availability of medication during the pandemic. GSIR framework can be utilized to find a good fit or predictive model for any pandemic. The study is performed on the COVID-19 data for various countries with detailed results for India, Brazil, United States of America (USA), and World. The peak infection, total expected number of COVID-19 cases and thereof deaths, time-varying reproduction number, and various other parameters are estimated from the available data using the proposed methodology. The proposed GSIR model advances the existing theory and yields promising results for continuous predictive monitoring of COVID-19 pandemic.
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Affiliation(s)
- Pushpendra Singh
- Department of Electronics & Communication Engineering, National Institute of Technology Hamirpur, Hamirpur, India.
| | - Anubha Gupta
- SBILab, Department of ECE, Indraprastha Institute of Information Technology Delhi, Delhi, India.
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8
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Cui K, KhudaBukhsh WR, Koeppl H. Motif-based mean-field approximation of interacting particles on clustered networks. Phys Rev E 2022; 105:L042301. [PMID: 35590665 DOI: 10.1103/physreve.105.l042301] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2022] [Accepted: 04/07/2022] [Indexed: 06/15/2023]
Abstract
Interacting particles on graphs are routinely used to study magnetic behavior in physics, disease spread in epidemiology, and opinion dynamics in social sciences. The literature on mean-field approximations of such systems for large graphs typically remains limited to specific dynamics, or assumes cluster-free graphs for which standard approximations based on degrees and pairs are often reasonably accurate. Here, we propose a motif-based mean-field approximation that considers higher-order subgraph structures in large clustered graphs. Numerically, our equations agree with stochastic simulations where existing methods fail.
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Affiliation(s)
- Kai Cui
- Department of Electrical Engineering and Information Technology, Technische Universität Darmstadt, 64287 Darmstadt, Germany
| | | | - Heinz Koeppl
- Department of Electrical Engineering and Information Technology, Technische Universität Darmstadt, 64287 Darmstadt, Germany
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9
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Aurell A, Carmona R, Dayanıklı G, Laurière M. Finite State Graphon Games with Applications to Epidemics. DYNAMIC GAMES AND APPLICATIONS 2022; 12:49-81. [PMID: 35003827 PMCID: PMC8720171 DOI: 10.1007/s13235-021-00410-2] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Accepted: 10/25/2021] [Indexed: 06/14/2023]
Abstract
We consider a game for a continuum of non-identical players evolving on a finite state space. Their heterogeneous interactions are represented with a graphon, which can be viewed as the limit of a dense random graph. A player's transition rates between the states depend on their control and the strength of interaction with the other players. We develop a rigorous mathematical framework for the game and analyze Nash equilibria. We provide a sufficient condition for a Nash equilibrium and prove existence of solutions to a continuum of fully coupled forward-backward ordinary differential equations characterizing Nash equilibria. Moreover, we propose a numerical approach based on machine learning methods and we present experimental results on different applications to compartmental models in epidemiology.
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Affiliation(s)
- Alexander Aurell
- Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544 USA
| | - René Carmona
- Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544 USA
| | - Gökçe Dayanıklı
- Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544 USA
| | - Mathieu Laurière
- Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544 USA
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10
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Otani Y, Kasai H, Tanigawara Y. Pharmacometric analysis of seasonal influenza epidemics and the effect of vaccination using sentinel surveillance data. CPT-PHARMACOMETRICS & SYSTEMS PHARMACOLOGY 2021; 11:44-54. [PMID: 34676676 PMCID: PMC8752114 DOI: 10.1002/psp4.12732] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/26/2021] [Revised: 09/26/2021] [Accepted: 10/01/2021] [Indexed: 11/28/2022]
Abstract
The identification of influenza epidemics and assessment of the efficacy of vaccination against this infection are major challenges for the implementation of effective public health strategies, such as vaccination programs. In this study, we developed a new pharmacometric model to evaluate the efficacy of vaccination based on infection surveillance data from the 2010/2011 to 2018/2019 influenza seasons in Japan. A novel susceptible‐infected‐removed plus vaccination model, based on an indirect response structure with the effect of vaccination, was applied to describe seasonal influenza epidemics using a preseasonal collection of data regarding serological H1 antibody titer positivity and the fraction of virus strains. Using this model, we evaluated Kin (a parameter describing the transmission rate of symptomatic influenza infection) for different age groups. Furthermore, we defined a new parameter (prevention factor) showing the efficacy of vaccination against each viral strain and in different age groups. We found that the prevention factor of vaccination against influenza varied among age groups. Notably, children aged 5–14 years showed the highest Kin value during the 10 influenza seasons and the greatest preventive effect of vaccination (prevention factor = 70.8%). The propagation of influenza epidemics varies in different age groups. Children aged 5–14 years most likely play a leading role in the transmission of influenza. Prioritized vaccination in this age group may be the most effective strategy for reducing the prevalence of influenza in the community.
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Affiliation(s)
- Yuki Otani
- Laboratory of Pharmacometrics and Systems Pharmacology, Keio Frontier Research and Education Collaboration Square at Tonomachi, Kanagawa, Japan.,Keio University Graduate School of Medicine, Tokyo, Japan
| | - Hidefumi Kasai
- Laboratory of Pharmacometrics and Systems Pharmacology, Keio Frontier Research and Education Collaboration Square at Tonomachi, Kanagawa, Japan
| | - Yusuke Tanigawara
- Laboratory of Pharmacometrics and Systems Pharmacology, Keio Frontier Research and Education Collaboration Square at Tonomachi, Kanagawa, Japan
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11
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Syrowatka A, Kuznetsova M, Alsubai A, Beckman AL, Bain PA, Craig KJT, Hu J, Jackson GP, Rhee K, Bates DW. Leveraging artificial intelligence for pandemic preparedness and response: a scoping review to identify key use cases. NPJ Digit Med 2021; 4:96. [PMID: 34112939 PMCID: PMC8192906 DOI: 10.1038/s41746-021-00459-8] [Citation(s) in RCA: 29] [Impact Index Per Article: 9.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/15/2020] [Accepted: 04/26/2021] [Indexed: 02/05/2023] Open
Abstract
Artificial intelligence (AI) represents a valuable tool that could be widely used to inform clinical and public health decision-making to effectively manage the impacts of a pandemic. The objective of this scoping review was to identify the key use cases for involving AI for pandemic preparedness and response from the peer-reviewed, preprint, and grey literature. The data synthesis had two parts: an in-depth review of studies that leveraged machine learning (ML) techniques and a limited review of studies that applied traditional modeling approaches. ML applications from the in-depth review were categorized into use cases related to public health and clinical practice, and narratively synthesized. One hundred eighty-three articles met the inclusion criteria for the in-depth review. Six key use cases were identified: forecasting infectious disease dynamics and effects of interventions; surveillance and outbreak detection; real-time monitoring of adherence to public health recommendations; real-time detection of influenza-like illness; triage and timely diagnosis of infections; and prognosis of illness and response to treatment. Data sources and types of ML that were useful varied by use case. The search identified 1167 articles that reported on traditional modeling approaches, which highlighted additional areas where ML could be leveraged for improving the accuracy of estimations or projections. Important ML-based solutions have been developed in response to pandemics, and particularly for COVID-19 but few were optimized for practical application early in the pandemic. These findings can support policymakers, clinicians, and other stakeholders in prioritizing research and development to support operationalization of AI for future pandemics.
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Affiliation(s)
- Ania Syrowatka
- Division of General Internal Medicine, Brigham and Women's Hospital, Boston, MA, USA.
- Harvard Medical School, Boston, MA, USA.
| | | | - Ava Alsubai
- Division of General Internal Medicine, Brigham and Women's Hospital, Boston, MA, USA
| | - Adam L Beckman
- Harvard Medical School, Boston, MA, USA
- Harvard Business School, Boston, MA, USA
| | - Paul A Bain
- Countway Library of Medicine, Harvard Medical School, Boston, MA, USA
| | | | - Jianying Hu
- IBM Research, Center for Computational Health, Yorktown Heights, NY, USA
| | - Gretchen Purcell Jackson
- IBM Watson Health, Cambridge, MA, USA
- Department of Pediatric Surgery, Vanderbilt University Medical Center, Nashville, TN, USA
| | - Kyu Rhee
- IBM Watson Health, Cambridge, MA, USA
- CVS Health, Wellesley Hills, MA, USA
| | - David W Bates
- Division of General Internal Medicine, Brigham and Women's Hospital, Boston, MA, USA
- Harvard Medical School, Boston, MA, USA
- Harvard T. H. Chan School of Public Health, Boston, MA, USA
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12
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Yasir KA, Liu WM. Social distancing mediated generalized model to predict epidemic spread of COVID-19. NONLINEAR DYNAMICS 2021; 106:1187-1195. [PMID: 33867677 PMCID: PMC8038536 DOI: 10.1007/s11071-021-06424-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/29/2020] [Accepted: 03/29/2021] [Indexed: 06/12/2023]
Abstract
The extensive proliferation of recent coronavirus (COVID-19), all over the world, is the outcome of social interactions through massive transportation, gatherings and population growth. To disrupt the widespread of COVID-19, a mechanism for social distancing is indispensable. Also, to predict the effectiveness and quantity of social distancing for a particular social network, with a certain contagion, a generalized model is needed. In this manuscript, we propose a social distancing mediated generalized model to predict the pandemic spread of COVID-19. By considering growth rate as a temporal harmonic function damped with social distancing in generalized Richard model and by using the data of confirmed COVID-19 cases in China, USA and India, we find that, with time, the cumulative spread grows more rapidly due to weak social distancing as compared to the stronger social distancing, where it is explicitly decreasing. Furthermore, we predict the possible outcomes with various social distancing scenarios by considering highest growth rate as an initial state, and illustrate that the increase in social distancing tremendously decreases growth rate, even it tends to reach zero in lockdown regimes. Our findings not only provide epidemic growth scenarios as a function of social distancing but also provide a modified growth model to predict controlled information flow in any network.
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Affiliation(s)
| | - Wu-Ming Liu
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, 100190 China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100190 China
- Songshan Lake Materials Laboratory, Dongguan, 523808 Guangdong China
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13
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McGahan I, Powell J, Spencer E. 28 Models Later: Model Competition and the Zombie Apocalypse. Bull Math Biol 2021; 83:22. [PMID: 33452943 PMCID: PMC7811353 DOI: 10.1007/s11538-020-00845-5] [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: 04/20/2020] [Accepted: 12/09/2020] [Indexed: 11/06/2022]
Abstract
Between Fall 2011 and Fall 2012 students at Utah State University played several rounds of Humans versus Zombies (HvZ), a role-playing variant of tag popular on college campuses. The goal of the game is for the zombies to tag humans, converting them into more zombies. Based on portrayals of 'zombieism' in popular culture, one might treat HvZ as a disease system. However, a traditional SIR model with mass-action dynamics does a poor job of modeling HvZ, leading to the natural question: What mechanisms drive the dynamics of the HvZ system? We use model competition, with Bayesian Information Criterion as arbiter, to answer this question. First, we develop a suite of models with a variety of transmission mechanisms and fit to data from fall 2011. We use model competition to determine which model(s) have the most support from the data, thereby offering insight into driving mechanisms for HvZ. Bootstrapping is used to both assess the significance of individual mechanisms and to determine confidence in the performance of our models. Finally, we test predictions of the best models with data from fall 2012. Results indicate that through both years of the game humans tend to cluster defensively, zombies tend to hunt in groups, some zombies are more proficient hunters, and some humans leave the game.
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Affiliation(s)
- Ian McGahan
- Department of Mathematics and Statistics, Utah State University, Logan, USA.
| | - James Powell
- Department of Mathematics and Statistics, Utah State University, Logan, USA
| | - Elizabeth Spencer
- Department of Statistics and Applied Probability, University of California Santa Barbara, Santa Barbara, USA
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14
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Sakaguchi H, Nakao Y. Slow decay of infection in the inhomogeneous susceptible-infected-recovered model. Phys Rev E 2021; 103:012301. [PMID: 33601590 DOI: 10.1103/physreve.103.012301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2020] [Accepted: 12/22/2020] [Indexed: 06/12/2023]
Abstract
The susceptible-infected-recovered (SIR) model with spatially inhomogeneous infection rate is studied with numerical simulations in one, two, and three dimensions, considering the case that the infection spreads inhomogeneously in densely populated regions or hot spots. We find that the total population of infection decays very slowly in the inhomogeneous systems in some cases, in contrast to the exponential decay of the infected population I(t) in the SIR model of the ordinary differential equation. The slow decay of the infected population suggests that the infection is locally maintained for long and it is difficult for the disease to disappear completely.
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Affiliation(s)
- Hidetsugu Sakaguchi
- Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
| | - Yuta Nakao
- Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
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15
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Leng A, Maitland E, Wang S, Nicholas S, Liu R, Wang J. Individual preferences for COVID-19 vaccination in China. Vaccine 2020; 39:247-254. [PMID: 33328140 PMCID: PMC7719001 DOI: 10.1016/j.vaccine.2020.12.009] [Citation(s) in RCA: 109] [Impact Index Per Article: 27.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/27/2020] [Revised: 11/29/2020] [Accepted: 12/02/2020] [Indexed: 01/07/2023]
Abstract
BACKGROUND Vaccinations are an effective choice to stop disease outbreaks, including COVID-19. There is little research on individuals' COVID-19 vaccination decision-making. OBJECTIVE We aimed to determine individual preferences for COVID-19 vaccinations in China, and to assess the factors influencing vaccination decision-making to facilitate vaccination coverage. METHODS A D-efficient discrete choice experiment was conducted across six Chinese provinces selected by the stratified random sampling method. Vaccine choice sets were constructed using seven attributes: vaccine effectiveness, side-effects, accessibility, number of doses, vaccination sites, duration of vaccine protection, and proportion of acquaintances vaccinated. Conditional logit and latent class models were used to identify preferences. RESULTS Although all seven attributes were proved to significantly influence respondents' vaccination decision, vaccine effectiveness, side-effects and proportion of acquaintances vaccinated were the most important. We also found a higher probability of vaccinating when the vaccine was more effective; risks of serious side effects were small; vaccinations were free and voluntary; the fewer the number of doses; the longer the protection duration; and the higher the proportion of acquaintances vaccinated. Higher local vaccine coverage created altruistic herd incentives to vaccinate rather than free-rider problems. The predicted vaccination uptake of the optimal vaccination scenario in our study was 84.77%. Preference heterogeneity was substantial. Individuals who were older, had a lower education level, lower income, higher trust in the vaccine and higher perceived risk of infection, displayed a higher probability to vaccinate. CONCLUSIONS Preference heterogeneity among individuals should lead health authorities to address the diversity of expectations about COVID-19 vaccinations. To maximize COVID-19 vaccine uptake, health authorities should promote vaccine effectiveness; pro-actively communicate the absence or presence of vaccine side effects; and ensure rapid and wide media communication about local vaccine coverage.
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Affiliation(s)
- Anli Leng
- School of Political Science and Public Administration, Institute of Governance, Shandong University, 72 Binhai Rd, Qingdao, Shandong 266237, China.
| | - Elizabeth Maitland
- School of Management, University of Liverpool, Chatham Building, Chatham Street, Liverpool L697ZH, England, United Kingdom.
| | - Siyuan Wang
- University of Melbourne, 369 Abbotsford Street, North Melbourne, VIC 3051, Australia.
| | - Stephen Nicholas
- Australian National Institute of Management and Commerce, 1 Central Avenue Australian Technology Park, Eveleigh Sydney, NSW 2015, Australia; Newcastle Business School, University of Newcastle, University Drive, Newcastle, NSW 2308, Australia.
| | - Rugang Liu
- School of Health Policy & Management, Nanjing Medical University, No. 101 Longmian Avenue, Jiangning District, Nanjing 211166, China.
| | - Jian Wang
- Dong Fureng Institute of Economic and Social Development, Wuhan University, 54 Dongsi Lishi Hutong, Beijing 100010, China; Center for Health Economics and Management at School of Economics and Management, Wuhan University, Wuhan, Hubei Province 430072, China.
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16
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Hong HG, Li Y. Estimation of time-varying reproduction numbers underlying epidemiological processes: A new statistical tool for the COVID-19 pandemic. PLoS One 2020; 15:e0236464. [PMID: 32692753 PMCID: PMC7373269 DOI: 10.1371/journal.pone.0236464] [Citation(s) in RCA: 54] [Impact Index Per Article: 13.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2020] [Accepted: 07/07/2020] [Indexed: 11/18/2022] Open
Abstract
The coronavirus pandemic has rapidly evolved into an unprecedented crisis. The susceptible-infectious-removed (SIR) model and its variants have been used for modeling the pandemic. However, time-independent parameters in the classical models may not capture the dynamic transmission and removal processes, governed by virus containment strategies taken at various phases of the epidemic. Moreover, few models account for possible inaccuracies of the reported cases. We propose a Poisson model with time-dependent transmission and removal rates to account for possible random errors in reporting and estimate a time-dependent disease reproduction number, which may reflect the effectiveness of virus control strategies. We apply our method to study the pandemic in several severely impacted countries, and analyze and forecast the evolving spread of the coronavirus. We have developed an interactive web application to facilitate readers' use of our method.
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Affiliation(s)
- Hyokyoung G. Hong
- Department of Statistics and Probability, Michigan State University, East Lansing, MI, United States of America
| | - Yi Li
- Department of Biostatistics, University of Michigan, Ann Arbor, MI, United States of America
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17
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Ezanno P, Andraud M, Beaunée G, Hoch T, Krebs S, Rault A, Touzeau S, Vergu E, Widgren S. How mechanistic modelling supports decision making for the control of enzootic infectious diseases. Epidemics 2020; 32:100398. [PMID: 32622313 DOI: 10.1016/j.epidem.2020.100398] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2019] [Revised: 05/07/2020] [Accepted: 05/29/2020] [Indexed: 12/28/2022] Open
Abstract
Controlling enzootic diseases, which generate a large cumulative burden and are often unregulated, is needed for sustainable farming, competitive agri-food chains, and veterinary public health. We discuss the benefits and challenges of mechanistic epidemiological modelling for livestock enzootics, with particular emphasis on the need for interdisciplinary approaches. We focus on issues arising when modelling pathogen spread at various scales (from farm to the region) to better assess disease control and propose targeted options. We discuss in particular the inclusion of farmers' strategic decision-making, the integration of within-host scale to refine intervention targeting, and the need to ground models on data.
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Affiliation(s)
- P Ezanno
- INRAE, Oniris, BIOEPAR, Site de la Chantrerie, CS40706, 44307 Nantes, France.
| | - M Andraud
- Unité épidémiologie et bien-être du porc, Anses Laboratoire de Ploufragan-Plouzané, Ploufragan, France.
| | - G Beaunée
- INRAE, Oniris, BIOEPAR, Site de la Chantrerie, CS40706, 44307 Nantes, France.
| | - T Hoch
- INRAE, Oniris, BIOEPAR, Site de la Chantrerie, CS40706, 44307 Nantes, France.
| | - S Krebs
- INRAE, Oniris, BIOEPAR, Site de la Chantrerie, CS40706, 44307 Nantes, France.
| | - A Rault
- INRAE, Oniris, BIOEPAR, Site de la Chantrerie, CS40706, 44307 Nantes, France.
| | - S Touzeau
- INRAE, CNRS, Université Côte d'Azur, ISA, France; Inria, INRAE, CNRS, Université Paris Sorbonne, Université Côte d'Azur, BIOCORE, France.
| | - E Vergu
- INRAE, Université Paris-Saclay, MaIAGE, 78350 Jouy-en-Josas, France.
| | - S Widgren
- Department of Disease Control and Epidemiology, National Veterinary Institute, 751 89 Uppsala, Sweden.
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18
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Wu K, Darcet D, Wang Q, Sornette D. Generalized logistic growth modeling of the COVID-19 outbreak: comparing the dynamics in the 29 provinces in China and in the rest of the world. NONLINEAR DYNAMICS 2020; 101:1561-1581. [PMID: 32836822 PMCID: PMC7437112 DOI: 10.1007/s11071-020-05862-6] [Citation(s) in RCA: 87] [Impact Index Per Article: 21.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2020] [Accepted: 07/29/2020] [Indexed: 05/04/2023]
Abstract
Started in Wuhan, China, the COVID-19 has been spreading all over the world. We calibrate the logistic growth model, the generalized logistic growth model, the generalized Richards model and the generalized growth model to the reported number of infected cases for the whole of China, 29 provinces in China, and 33 countries and regions that have been or are undergoing major outbreaks. We dissect the development of the epidemics in China and the impact of the drastic control measures both at the aggregate level and within each province. We quantitatively document four phases of the outbreak in China with a detailed analysis on the heterogeneous situations across provinces. The extreme containment measures implemented by China were very effective with some instructive variations across provinces. Borrowing from the experience of China, we made scenario projections on the development of the outbreak in other countries. We identified that outbreaks in 14 countries (mostly in western Europe) have ended, while resurgences of cases have been identified in several among them. The modeling results clearly show longer after-peak trajectories in western countries, in contrast to most provinces in China where the after-peak trajectory is characterized by a much faster decay. We identified three groups of countries in different level of outbreak progress, and provide informative implications for the current global pandemic.
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Affiliation(s)
- Ke Wu
- Institute of Risk Analysis, Prediction and Management (Risks-X), Academy for Advanced Interdisciplinary Studies, Southern University of Science and Technology (SUSTech), Shenzhen, China
- Department of Management, Technology and Economics (D-MTEC), Chair of Entrepreneurial Risks, ETH Zurich, Zurich, Switzerland
| | | | - Qian Wang
- Institute of Risk Analysis, Prediction and Management (Risks-X), Academy for Advanced Interdisciplinary Studies, Southern University of Science and Technology (SUSTech), Shenzhen, China
- Department of Management, Technology and Economics (D-MTEC), Chair of Entrepreneurial Risks, ETH Zurich, Zurich, Switzerland
- Department of Banking and Finance, University of Zurich, Zurich, Switzerland
| | - Didier Sornette
- Institute of Risk Analysis, Prediction and Management (Risks-X), Academy for Advanced Interdisciplinary Studies, Southern University of Science and Technology (SUSTech), Shenzhen, China
- Department of Management, Technology and Economics (D-MTEC), Chair of Entrepreneurial Risks, ETH Zurich, Zurich, Switzerland
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19
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Yan J, Guha S, Hariharan P, Myers M. Modeling the Effectiveness of Respiratory Protective Devices in Reducing Influenza Outbreak. RISK ANALYSIS : AN OFFICIAL PUBLICATION OF THE SOCIETY FOR RISK ANALYSIS 2019; 39:647-661. [PMID: 30229968 DOI: 10.1111/risa.13181] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/10/2017] [Revised: 07/05/2018] [Accepted: 07/10/2018] [Indexed: 05/04/2023]
Abstract
Outbreaks of influenza represent an important health concern worldwide. In many cases, vaccines are only partially successful in reducing the infection rate, and respiratory protective devices (RPDs) are used as a complementary countermeasure. In devising a protection strategy against influenza for a given population, estimates of the level of protection afforded by different RPDs is valuable. In this article, a risk assessment model previously developed in general form was used to estimate the effectiveness of different types of protective equipment in reducing the rate of infection in an influenza outbreak. It was found that a 50% compliance in donning the device resulted in a significant (at least 50% prevalence and 20% cumulative incidence) reduction in risk for fitted and unfitted N95 respirators, high-filtration surgical masks, and both low-filtration and high-filtration pediatric masks. An 80% compliance rate essentially eliminated the influenza outbreak. The results of the present study, as well as the application of the model to related influenza scenarios, are potentially useful to public health officials in decisions involving resource allocation or education strategies.
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Affiliation(s)
- Jing Yan
- Departments of Chemical & Biomolecular Engineering, University of Maryland, College Park, MD, USA
- Division of Applied Mechanics, Office of Science and Engineering Laboratories, Center for Devices and Radiological Health, Food and Drug Administration, Silver Spring, MD, USA
| | - Suvajyoti Guha
- Division of Applied Mechanics, Office of Science and Engineering Laboratories, Center for Devices and Radiological Health, Food and Drug Administration, Silver Spring, MD, USA
| | - Prasanna Hariharan
- Division of Applied Mechanics, Office of Science and Engineering Laboratories, Center for Devices and Radiological Health, Food and Drug Administration, Silver Spring, MD, USA
| | - Matthew Myers
- Division of Applied Mechanics, Office of Science and Engineering Laboratories, Center for Devices and Radiological Health, Food and Drug Administration, Silver Spring, MD, USA
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20
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Optimal time-profiles of public health intervention to shape voluntary vaccination for childhood diseases. J Math Biol 2018; 78:1089-1113. [PMID: 30390103 DOI: 10.1007/s00285-018-1303-1] [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: 09/18/2017] [Revised: 09/10/2018] [Indexed: 10/27/2022]
Abstract
In order to seek the optimal time-profiles of public health systems (PHS) Intervention to favor vaccine propensity, we apply optimal control (OC) to a SIR model with voluntary vaccination and PHS intervention. We focus on short-term horizons, and on both continuous control strategies resulting from the forward-backward sweep deterministic algorithm, and piecewise-constant strategies (which are closer to the PHS way of working) investigated by the simulated annealing (SA) stochastic algorithm. For childhood diseases, where disease costs are much larger than vaccination costs, the OC solution sets at its maximum for most of the policy horizon, meaning that the PHS cannot further improve perceptions about the net benefit of immunization. Thus, the subsequent dynamics of vaccine uptake stems entirely from the declining perceived risk of infection (due to declining prevalence) which is communicated by direct contacts among parents, and unavoidably yields a future decline in vaccine uptake. We find that for relatively low communication costs, the piecewise control is close to the continuous control. For large communication costs the SA algorithm converges towards a non-monotone OC that can have oscillations.
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21
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Individual decisions to vaccinate one's child or oneself: A discrete choice experiment rejecting free-riding motives. Soc Sci Med 2018; 207:106-116. [PMID: 29738898 DOI: 10.1016/j.socscimed.2018.04.038] [Citation(s) in RCA: 39] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/11/2017] [Revised: 04/18/2018] [Accepted: 04/24/2018] [Indexed: 02/03/2023]
Abstract
It is essential for public health to understand what drives people's hesitance towards vaccination. Theoretical models of vaccination decisions are ubiquitous, often incorporating herd immunity, perceptions of vaccine-related side-effects (VRSE) and of vaccine-preventable burden of disease, but with little to no empirical exploration. Herd immunity is a (usually) positive externality where vaccinated individuals influence others' risks by their reduced capability to transmit an infectious disease to them. It is often assumed that (rational) individuals incorporate this externality in their strategic vaccination decision, from which free-riding behavior arises. We performed a Bayesian D-efficient discrete choice experiment in February-March 2017 to study vaccination behavior in 1919 Belgian respondents. Choice sets with vaccine profiles were constructed using six attributes: vaccine effectiveness, VRSE, accessibility (in terms of convenience and reimbursement), vaccine-preventable burden of disease, local (respondents' network of contacts) vaccination coverage, and population (the population at large) vaccination coverage. VRSE and accessibility are the most influential attributes, followed by vaccine effectiveness and burden of disease. Both population and local coverage are less important than the other attributes, but show a significant direct linear relationship with vaccine utility. This supports the existence of peer influence (more incentivized as more and more vaccinate), rather than free-riding on herd immunity. These findings were independent of whether respondents made vaccine choices for themselves or for their child. Around 40% of the respondents indicated accepting vaccination with little or no questioning. These 'acceptors' were less sensitive to changes in the vaccine-preventable burden of disease for their child's vaccination choices (but not for themselves). Public health institutions are critical in stimulating vaccine uptake by making vaccines conveniently available at an affordable price and by communicating pro-actively on perceived VRSEs. The free-riding assumption as a driver of individual vaccine decisions, seems inappropriate, but this observation needs confirming in other populations.
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22
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Canonical modeling of anticipatory vaccination behavior and long term epidemic recurrence. J Theor Biol 2018; 436:26-38. [DOI: 10.1016/j.jtbi.2017.09.023] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/10/2017] [Revised: 05/30/2017] [Accepted: 09/21/2017] [Indexed: 11/23/2022]
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23
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Verelst F, Willem L, Beutels P. Behavioural change models for infectious disease transmission: a systematic review (2010-2015). J R Soc Interface 2016; 13:20160820. [PMID: 28003528 PMCID: PMC5221530 DOI: 10.1098/rsif.2016.0820] [Citation(s) in RCA: 169] [Impact Index Per Article: 21.1] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2016] [Accepted: 11/25/2016] [Indexed: 12/13/2022] Open
Abstract
We review behavioural change models (BCMs) for infectious disease transmission in humans. Following the Cochrane collaboration guidelines and the PRISMA statement, our systematic search and selection yielded 178 papers covering the period 2010-2015. We observe an increasing trend in published BCMs, frequently coupled to (re)emergence events, and propose a categorization by distinguishing how information translates into preventive actions. Behaviour is usually captured by introducing information as a dynamic parameter (76/178) or by introducing an economic objective function, either with (26/178) or without (37/178) imitation. Approaches using information thresholds (29/178) and exogenous behaviour formation (16/178) are also popular. We further classify according to disease, prevention measure, transmission model (with 81/178 population, 6/178 metapopulation and 91/178 individual-level models) and the way prevention impacts transmission. We highlight the minority (15%) of studies that use any real-life data for parametrization or validation and note that BCMs increasingly use social media data and generally incorporate multiple sources of information (16/178), multiple types of information (17/178) or both (9/178). We conclude that individual-level models are increasingly used and useful to model behaviour changes. Despite recent advancements, we remain concerned that most models are purely theoretical and lack representative data and a validation process.
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Affiliation(s)
- Frederik Verelst
- Centre for Health Economics Research and Modelling Infectious Diseases, Vaccine and Infectious Disease Institute, University of Antwerp, Antwerp, Belgium
| | - Lander Willem
- Centre for Health Economics Research and Modelling Infectious Diseases, Vaccine and Infectious Disease Institute, University of Antwerp, Antwerp, Belgium
| | - Philippe Beutels
- Centre for Health Economics Research and Modelling Infectious Diseases, Vaccine and Infectious Disease Institute, University of Antwerp, Antwerp, Belgium
- School of Public Health and Community Medicine, The University of New South Wales, Sydney, New South Wales, Australia
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24
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Doutor P, Rodrigues P, Soares MDC, Chalub FACC. Optimal vaccination strategies and rational behaviour in seasonal epidemics. J Math Biol 2016; 73:1437-1465. [DOI: 10.1007/s00285-016-0997-1] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/09/2015] [Revised: 03/17/2016] [Indexed: 01/13/2023]
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25
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Swiecicki I, Gobron T, Ullmo D. Schrödinger Approach to Mean Field Games. PHYSICAL REVIEW LETTERS 2016; 116:128701. [PMID: 27058107 DOI: 10.1103/physrevlett.116.128701] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2015] [Indexed: 06/05/2023]
Abstract
Mean field games (MFG) provide a theoretical frame to model socioeconomic systems. In this Letter, we study a particular class of MFG that shows strong analogies with the nonlinear Schrödinger and Gross-Pitaevskii equations introduced in physics to describe a variety of physical phenomena. Using this bridge, many results and techniques developed along the years in the latter context can be transferred to the former, which provides both a new domain of application for the nonlinear Schrödinger equation and a new and fruitful approach in the study of mean field games. Utilizing this approach, we analyze in detail a population dynamics model in which the "players" are under a strong incentive to coordinate themselves.
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Affiliation(s)
- Igor Swiecicki
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
- LPTM, CNRS, Université Cergy-Pontoise, 95302 Cergy-Pontoise, France
| | - Thierry Gobron
- LPTM, CNRS, Université Cergy-Pontoise, 95302 Cergy-Pontoise, France
| | - Denis Ullmo
- LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
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