1
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Zang H, Liu S, Lin Y. Evaluations of heterogeneous epidemic models with exponential and non-exponential distributions for latent period: the Case of COVID-19. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:12579-12598. [PMID: 37501456 DOI: 10.3934/mbe.2023560] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 07/29/2023]
Abstract
Most of heterogeneous epidemic models assume exponentially distributed sojourn times in infectious states, which may not be practical in reality and could affect the dynamics of the epidemic. This paper investigates the potential discrepancies between exponential and non-exponential distribution models in analyzing the transmission patterns of infectious diseases and evaluating control measures. Two SEIHR models with multiple subgroups based on different assumptions for latency are established: Model Ⅰ assumes an exponential distribution of latency, while Model Ⅱ assumes a gamma distribution. To overcome the challenges associated with the high dimensionality of GDM, we derive the basic reproduction number ($ R_{0} $) of the model theoretically, and apply numerical simulations to evaluate the effect of different interventions on EDM and GDM. Our results show that considering a more realistic gamma distribution of latency can change the peak numbers of infected and the timescales of an epidemic, and GDM may underestimate the infection eradication time and overestimate the peak value compared to EDM. Additionally, the two models can produce inconsistent predictions in estimating the time to reach the peak. Our study contributes to a more accurate understanding of disease transmission patterns, which is crucial for effective disease control and prevention.
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Affiliation(s)
- Huiping Zang
- School of Mathematical Sciences, Tiangong University, Tianjin 300387, China
| | - Shengqiang Liu
- School of Mathematical Sciences, Tiangong University, Tianjin 300387, China
| | - Yi Lin
- School of Mathematical Sciences, Tiangong University, Tianjin 300387, China
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2
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Maity B, Banerjee S, Senapati A, Chattopadhyay J. Quantifying optimal resource allocation strategies for controlling epidemics. J R Soc Interface 2023; 20:20230036. [PMID: 37194270 PMCID: PMC10189312 DOI: 10.1098/rsif.2023.0036] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2023] [Accepted: 04/25/2023] [Indexed: 05/18/2023] Open
Abstract
Frequent emergence of communicable diseases is a major concern worldwide. Lack of sufficient resources to mitigate the disease burden makes the situation even more challenging for lower-income countries. Hence, strategy development for disease eradication and optimal management of the social and economic burden has garnered a lot of attention in recent years. In this context, we quantify the optimal fraction of resources that can be allocated to two major intervention measures, namely reduction of disease transmission and improvement of healthcare infrastructure. Our results demonstrate that the effectiveness of each of the interventions has a significant impact on the optimal resource allocation in both long-term disease dynamics and outbreak scenarios. The optimal allocation strategy for long-term dynamics exhibits non-monotonic behaviour with respect to the effectiveness of interventions, which differs from the more intuitive strategy recommended in the case of outbreaks. Further, our results indicate that the relationship between investment in interventions and the corresponding increase in patient recovery rate or decrease in disease transmission rate plays a decisive role in determining optimal strategies. Intervention programmes with decreasing returns promote the necessity for resource sharing. Our study provides fundamental insights into determining the best response strategy when controlling epidemics in resource-constrained situations.
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Affiliation(s)
- Biplab Maity
- Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India
| | - Swarnendu Banerjee
- Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India
- Copernicus Institute of Sustainable Development, Utrecht University, PO Box 80115, Utrecht 3508 TC, The Netherlands
| | - Abhishek Senapati
- Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India
- Center for Advanced Systems Understanding (CASUS), Untermarkt 20, Goerlitz 02826, Germany
| | - Joydev Chattopadhyay
- Agricultural and Ecological Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata 700108, India
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3
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Ma R, Shi L, Sun G. Policy Disparities Between Singapore and Israel in Response to the First Omicron Wave. Risk Manag Healthc Policy 2023; 16:489-502. [PMID: 37035268 PMCID: PMC10078824 DOI: 10.2147/rmhp.s402813] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Accepted: 03/14/2023] [Indexed: 04/04/2023] Open
Abstract
Purpose The purpose of this study is to evaluate public health measures during the first Omicron wave in Singapore and Israel to inform other countries confronted by COVID-19 outbreaks. Methods A comparative analysis was conducted using epidemiological data from Singapore and Israel between November 25th, 2021 and May 2nd, 2022 and policy information to examine the effects of public health measures in the two countries during the COVID-19 pandemic. Results Public health measures implemented by Singapore and Israel in response to the first Omicron wave were primarily intended to mitigate the effects of the COVID-19 pandemic. In Singapore, the pandemic led to more than 910,000 confirmed cases, a mortality rate of approximately 0.047%, a hospitalization rate of approximately 10.95%, and a severe illness rate of approximately 0.48%, without a second peak. In Israel, the pandemic not only resulted in over 2.74 million confirmed cases, a mortality rate of 0.095%, a hospitalization rate of about 7.39%, and a severe illness rate of approximately 2.30% but also returned after the significant relaxation of prevention regulations from March 1st, 2022. Conclusion Early and strict border control measures and surveillance measures are more effective in preventing and controlling the rapid spread of new strains of COVID-19 in the early stage. Furthermore, to prevent and control this highly infectious disease, COVID-19 vaccinations and booster shots must be promoted as soon as possible, medical service capacity must be enhanced, the hierarchical medical system must be improved, and non-pharmacological interventions must be implemented.
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Affiliation(s)
- Rongcai Ma
- Department of Health Management, School of Health Management, Southern Medical University, Guangzhou, Guangdong, 510515, People’s Republic of China
| | - Leiyu Shi
- Department of Health Policy and Management, Bloomberg School of Public Health, Johns Hopkins University, Baltimore, MD, 21205, USA
| | - Gang Sun
- Department of Health Management, School of Health Management, Southern Medical University, Guangzhou, Guangdong, 510515, People’s Republic of China
- Department of Health Policy and Management, Bloomberg School of Public Health, Johns Hopkins University, Baltimore, MD, 21205, USA
- Correspondence: Gang Sun, Department of Health Management, School of Health Management, Southern Medical University, Guangzhou, Guangdong, 510515, People’s Republic of China, Email
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4
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Fierro R. Cumulative damage for multi-type epidemics and an application to infectious diseases. J Math Biol 2023; 86:47. [PMID: 36797526 PMCID: PMC9934514 DOI: 10.1007/s00285-023-01880-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2022] [Revised: 12/04/2022] [Accepted: 01/23/2023] [Indexed: 02/18/2023]
Abstract
A continuous time multivariate stochastic model is proposed for assessing the damage of a multi-type epidemic cause to a population as it unfolds. The instants when cases occur and the magnitude of their injure are random. Thus, we define a cumulative damage based on counting processes and a multivariate mark process. For a large population we approximate the behavior of this damage process by its asymptotic distribution. Also, we analyze the distribution of the stopping times when the numbers of cases caused by the epidemic attain levels beyond certain thresholds. We focus on introducing some tools for statistical inference on the parameters related with the epidemic. In this regard, we present a general hypothesis test for homogeneity in epidemics and apply it to data of Covid-19 in Chile.
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Affiliation(s)
- Raúl Fierro
- Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile.
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5
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Serra M, Al-Mosleh S, Prasath G, Raju V, Mantena S, Chandra J, Iams S, Mahadevan L. Optimal policies for mitigating pandemic costs: a minimal model. Phys Biol 2022; 19. [PMID: 35790172 DOI: 10.1088/1478-3975/ac7e9e] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2021] [Accepted: 07/05/2022] [Indexed: 11/11/2022]
Abstract
There have been a number of pharmaceutical and non-pharmaceutical interventions associated with COVID-19 over the past two years. Of the various non-pharmaceutical interventions that were proposed and implemented to control the spread of the COVID-19 pandemic partial and complete lockdowns were used repeatedly in an attempt to minimize the costs associated with mortality, economic losses and social factors, while being subject to constraints such as finite hospital capacity. Here, we use a minimal model to understand the costs and benefits of these strategies that mitigate pandemic costs subject to constraints, we adopt the language of optimal control theory. This allows us to determine top-down policies for the nature and dynamics of social contact rates given an age-structured model for the dynamics of the disease. Depending on the relative weights allocated to mortality and socioeconomic losses, we see that the optimal strategies range from long-term social-distancing only for the most vulnerable, to partial lockdown to ensure not over-running hospitals, to alternating-shifts with significant reduction in mortality and/or socioeconomic losses. Crucially, commonly used strategies that involve long periods of broad lockdown are almost never optimal, as they are highly unstable to reopening {and entail high socioeconomic costs}. Using parameter estimates from data available for Germany and the USA early in the pandemic, we quantify these policies and use sensitivity analysis in the relevant model parameters and initial conditions to determine the range of robustness of our policies. Finally we also discuss how bottom-up behavioral changes affect the dynamics of the pandemic and show they can work in tandem with top-down control policies to mitigate pandemic costs even more effectively.
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Affiliation(s)
- Mattia Serra
- Harvard University, Pierce Hall, Cambridge, Cambridge, 02138, UNITED STATES
| | - Salem Al-Mosleh
- School of Engineering and Applied Sciences, Harvard University, Pierce Hall, Cambridge, Cambridge, 02138, UNITED STATES
| | - Ganga Prasath
- School of Engineering and Applied Sciences, Harvard University, 29 Oxford St, Cambridge, Cambridge, Massachusetts, 02138, UNITED STATES
| | - Vidya Raju
- School of Engineering and Applied Sciences, Harvard University, 29 Oxford St, Cambridge, Cambridge, Massachusetts, 02138, UNITED STATES
| | - Sreekar Mantena
- School of Engineering and Applied Sciences, Harvard University, 29 Oxford St, Cambridge, Cambridge, 02138, UNITED STATES
| | - Jay Chandra
- School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, 02138, UNITED STATES
| | - Sarah Iams
- Harvard University, 29 Oxford Street, Cambridge, 02138, UNITED STATES
| | - L Mahadevan
- Harvard University, 29 Oxford Street, Cambridge, Massachusetts, 02138, UNITED STATES
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6
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Rigatos G, Abbaszadeh M, Cuccurullo G. A nonlinear optimal control method against the spreading of epidemics. INT J BIOMATH 2022. [DOI: 10.1142/s1793524522500267] [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
To define a vaccination policy and antiviral treatment against the spreading of viral infections a nonlinear optimal (H-infinity) control approach is proposed. Actually, because of the scarcity of the resources for treating infectious diseases in terms of vaccines, antiviral drugs and other medical facilities, there is need to implement optimal control against the epidemics deployment. In this approach, the state-space model of the epidemics dynamics undergoes first approximate linearization around a temporary operating point which is recomputed at each time-step of the control method. The linearization is based on Taylor series expansion and on the computation of the associated Jacobian matrices. Next, an optimal (H-infinity) feedback controller is developed for the approximately linearized model of the epidemics. To compute the controller’s feedback gains an algebraic Riccati equation is solved at each iteration of the control algorithm. Furthermore, the global asymptotic stability properties of the control scheme are proven through Lyapunov stability analysis. This paper’s results confirm that optimal control of the infectious disease dynamics allows for eliminating its spreading while also keeping moderate the consumption of the related medication, that is vaccines and antiviral drugs.
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Affiliation(s)
- G. Rigatos
- Unit of Industrial Automation, Industrial Systems Institute, 26504, Rion Patras Greece, Greece
| | - M. Abbaszadeh
- Department of ECSE, Rensselaer Polytechnic Institute 12065, NY, USA
| | - G. Cuccurullo
- Department of Industrial Engineering, University of Salerno, Fisciano, 84084, Italy
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7
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Ma H, Zhang Q. Threshold dynamics and optimal control on an age-structured SIRS epidemic model with vaccination. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2021; 18:9474-9495. [PMID: 34814354 DOI: 10.3934/mbe.2021465] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
We consider a vaccination control into a age-structured susceptible-infective-recovered-susceptible (SIRS) model and study the global stability of the endemic equilibrium by the iterative method. The basic reproduction number $ R_0 $ is obtained. It is shown that if $ R_0 < 1 $, then the disease-free equilibrium is globally asymptotically stable, if $ R_0 > 1 $, then the disease-free and endemic equilibrium coexist simultaneously, and the global asymptotic stability of endemic equilibrium is also shown. Additionally, the Hamilton-Jacobi-Bellman (HJB) equation is given by employing the Bellman's principle of optimality. Through proving the existence of viscosity solution for HJB equation, we obtain the optimal vaccination control strategy. Finally, numerical simulations are performed to illustrate the corresponding analytical results.
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Affiliation(s)
- Han Ma
- School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China
| | - Qimin Zhang
- School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China
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8
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On the optimal control of SIR model with Erlang-distributed infectious period: isolation strategies. J Math Biol 2021; 83:36. [PMID: 34550465 PMCID: PMC8456197 DOI: 10.1007/s00285-021-01668-1] [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: 03/18/2021] [Revised: 07/22/2021] [Accepted: 09/08/2021] [Indexed: 11/15/2022]
Abstract
Mathematical models are formal and simplified representations of the knowledge related to a phenomenon. In classical epidemic models, a major simplification consists in assuming that the infectious period is exponentially distributed, then implying that the chance of recovery is independent on the time since infection. Here, we first attempt to investigate the consequences of relaxing this assumption on the performances of time-variant disease control strategies by using optimal control theory. In the framework of a basic susceptible–infected–removed (SIR) model, an Erlang distribution of the infectious period is considered and optimal isolation strategies are searched for. The objective functional to be minimized takes into account the cost of the isolation efforts per time unit and the sanitary costs due to the incidence of the epidemic outbreak. Applying the Pontryagin’s minimum principle, we prove that the optimal control problem admits only bang–bang solutions with at most two switches. In particular, the optimal strategy could be postponing the starting intervention time with respect to the beginning of the outbreak. Finally, by means of numerical simulations, we show how the shape of the optimal solutions is affected by the different distributions of the infectious period, by the relative weight of the two cost components, and by the initial conditions.
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9
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Bliman PA, Duprez M, Privat Y, Vauchelet N. Optimal Immunity Control and Final Size Minimization by Social Distancing for the SIR Epidemic Model. JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2021; 189:408-436. [PMID: 33678904 PMCID: PMC7918002 DOI: 10.1007/s10957-021-01830-1] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/24/2020] [Accepted: 02/02/2021] [Indexed: 05/06/2023]
Abstract
The aim of this article is to understand how to apply partial or total containment to SIR epidemic model during a given finite time interval in order to minimize the epidemic final size, that is the cumulative number of cases infected during the complete course of an epidemic. The existence and uniqueness of an optimal strategy are proved for this infinite-horizon problem, and a full characterization of the solution is provided. The best policy consists in applying the maximal allowed social distancing effort until the end of the interval, starting at a date that is not always the closest date and may be found by a simple algorithm. Both theoretical results and numerical simulations demonstrate that it leads to a significant decrease in the epidemic final size. We show that in any case the optimal intervention has to begin before the number of susceptible cases has crossed the herd immunity level, and that its limit is always smaller than this threshold. This problem is also shown to be equivalent to the minimum containment time necessary to stop at a given distance after this threshold value.
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Affiliation(s)
- Pierre-Alexandre Bliman
- Inria, Sorbonne Université, Université Paris-Diderot SPC, CNRS, Laboratoire Jacques-Louis Lions, équipe Mamba, Paris, France
| | - Michel Duprez
- Inria, équipe MIMESiS, Université de Strasbourg, ICUBE, équipe MLMS, Strasbourg, France
| | - Yannick Privat
- Université de Strasbourg, CNRS UMR 7501, INRIA, Institut de Recherche Mathématique Avancée (IRMA), 7 rue René Descartes, 67084 Strasbourg, France
| | - Nicolas Vauchelet
- LAGA, UMR 7539, CNRS, Université Sorbonne Paris Nord, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France
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10
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Náraigh LÓ, Byrne Á. Piecewise-constant optimal control strategies for controlling the outbreak of COVID-19 in the Irish population. Math Biosci 2020; 330:108496. [PMID: 33075364 PMCID: PMC7566875 DOI: 10.1016/j.mbs.2020.108496] [Citation(s) in RCA: 19] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2020] [Revised: 09/10/2020] [Accepted: 10/08/2020] [Indexed: 12/13/2022]
Abstract
We introduce a deterministic SEIR model and fit it to epidemiological data for the COVID-19 outbreak in Ireland. We couple the model to economic considerations - we formulate an optimal control problem in which the cost to the economy of the various non-pharmaceutical interventions is minimized, subject to hospital admissions never exceeding a threshold value corresponding to health-service capacity. Within the framework of the model, the optimal strategy of disease control is revealed to be one of disease suppression, rather than disease mitigation.
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Affiliation(s)
- Lennon Ó Náraigh
- School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland.
| | - Áine Byrne
- School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
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11
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Kantner M, Koprucki T. Beyond just "flattening the curve": Optimal control of epidemics with purely non-pharmaceutical interventions. JOURNAL OF MATHEMATICS IN INDUSTRY 2020; 10:23. [PMID: 32834921 DOI: 10.1186/s13362-020-0069-4] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2020] [Accepted: 08/06/2020] [Indexed: 05/24/2023]
Abstract
When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple "flattening of the curve". Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socio-economic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany.
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Affiliation(s)
- Markus Kantner
- Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, Berlin, 10117 Germany
| | - Thomas Koprucki
- Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, Berlin, 10117 Germany
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12
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Boujallal L, Elhia M, Balatif O. A novel control set-valued approach with application to epidemic models. JOURNAL OF APPLIED MATHEMATICS & COMPUTING 2020; 65:295-319. [PMID: 32837465 PMCID: PMC7355539 DOI: 10.1007/s12190-020-01392-x] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 04/20/2020] [Revised: 06/23/2020] [Accepted: 06/27/2020] [Indexed: 06/11/2023]
Abstract
This work is considered in the framework of studies dedicated to the control problems, especially in epidemiology where the scientist are concerned to develop effective control strategies to minimize the number of infected individuals. In this paper, we set this problem as an asymptotic target control problem under mixed state-control constraints, for a general class of ordinary differential equations that model the temporal evolution of disease spread. The set of initial data, from which the number of infected people decrease to zero, is generated by a new type of Lyapunov functions defined in the sense of viability theory. The associated controls are provided via selections of adequately designed feedback map. The existence of such selections is improved by using Micheal selection theorem. Finally, an application to the SIRS epidemic model, with numerical simulations, is given to show the efficiency of our approach. To the best of our knowledge, our work is the first one that used a set-valued approach based on the viability theory to deal with an epidemic control problem.
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Affiliation(s)
- Lahoucine Boujallal
- Department of Mathematics, Hassan II University, P.O. Box 5366, Casablanca, Morocco
| | - Mohamed Elhia
- MAEGE Laboratory, Hassan II University, Casablanca, Morocco
| | - Omar Balatif
- Dynamical Systems Laboratory, Mathematical Engineering Team, Chouaib Doukkali University, El Jadida, Morocco
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13
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Amaral MA, Dantas WG, Arenzon JJ. Skepticism and rumor spreading: The role of spatial correlations. Phys Rev E 2020; 101:062418. [PMID: 32688484 DOI: 10.1103/physreve.101.062418] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2020] [Accepted: 06/05/2020] [Indexed: 12/21/2022]
Abstract
Critical thinking and skepticism are fundamental mechanisms that one may use to prevent the spreading of rumors, fake news, and misinformation. We consider a simple model in which agents without previous contact with the rumor, being skeptically oriented, may convince spreaders to stop their activity or, once exposed to the rumor, decide not to propagate it as a consequence, for example, of fact checking. We extend a previous, mean-field analysis of the combined effect of these two mechanisms, active and passive skepticism, to include spatial correlations. This can be done either analytically, through the pair approximation, or simulating an agent-based version on diverse networks. Our results show that while in mean field there is no coexistence between spreaders and susceptibles (although, depending on the parameters, there may be bistability depending on the initial conditions), when spatial correlations are included, because of the protective effect of the isolation provided by removed agents, coexistence is possible.
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Affiliation(s)
- Marco Antonio Amaral
- Instituto de Humanidades, Artes e Ciências, Universidade Federal do Sul da Bahia, CEP 45638-000 Teixeira de Freitas, Bahia, Brazil
| | - W G Dantas
- Departamento de Ciências Exatas, EEIMVR, Universidade Federal Fluminense, CEP 27255-125, Volta Redonda, Rio de Janeiro, Brazil.,Instituto de Física, Universidade Federal do Rio Grande do Sul, CEP 91501-970 Porto Alegre, Rio Grande do Sul, Brazil
| | - Jeferson J Arenzon
- Instituto de Física, Universidade Federal do Rio Grande do Sul, CEP 91501-970 Porto Alegre, Rio Grande do Sul, Brazil.,Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos, Rio de Janeiro, Rio de Janeiro, Brazil
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14
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Newbold SC, Finnoff D, Thunström L, Ashworth M, Shogren JF. Effects of Physical Distancing to Control COVID-19 on Public Health, the Economy, and the Environment. ENVIRONMENTAL & RESOURCE ECONOMICS 2020; 76:705-729. [PMID: 32836854 PMCID: PMC7399603 DOI: 10.1007/s10640-020-00440-1] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 07/02/2020] [Indexed: 05/22/2023]
Abstract
Physical distancing measures are important tools to control disease spread, especially in the absence of treatments and vaccines. While distancing measures can safeguard public health, they also can profoundly impact the economy and may have important indirect effects on the environment. The extent to which physical distancing measures should be applied therefore depends on the trade-offs between their health benefits and their economic costs. We develop an epidemiological-economic model to examine the optimal duration and intensity of physical distancing measures aimed to control the spread of COVID-19. In an application to the United States, our model considers the trade-off between the lives saved by physical distancing-both directly from stemming the spread of the virus and indirectly from reductions in air pollution during the period of physical distancing-and the short- and long-run economic costs that ensue from such measures. We examine the effect of air pollution co-benefits on the optimal physical distancing policy and conduct sensitivity analyses to gauge the influence of several key parameters and uncertain model assumptions. Using recent estimates of the association between airborne particulate matter and the virulence of COVID-19, we find that accounting for air pollution co-benefits can significantly increase the intensity and duration of the optimal physical distancing policy. To conclude, we broaden our discussion to consider the possibility of durable changes in peoples' behavior that could alter local markets, the global economy, and our relationship to nature for years to come.
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Affiliation(s)
| | - David Finnoff
- Department of Economics, University of Wyoming, Laramie, WY USA
| | - Linda Thunström
- Department of Economics, University of Wyoming, Laramie, WY USA
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15
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Kantner M, Koprucki T. Beyond just "flattening the curve": Optimal control of epidemics with purely non-pharmaceutical interventions. JOURNAL OF MATHEMATICS IN INDUSTRY 2020; 10:23. [PMID: 32834921 PMCID: PMC7432561 DOI: 10.1186/s13362-020-00091-3] [Citation(s) in RCA: 25] [Impact Index Per Article: 6.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2020] [Accepted: 08/06/2020] [Indexed: 05/20/2023]
Abstract
When effective medical treatment and vaccination are not available, non-pharmaceutical interventions such as social distancing, home quarantine and far-reaching shutdown of public life are the only available strategies to prevent the spread of epidemics. Based on an extended SEIR (susceptible-exposed-infectious-recovered) model and continuous-time optimal control theory, we compute the optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found and complete containment (eradication of the epidemic) is impossible. In this case, the optimal control must meet competing requirements: First, the minimization of disease-related deaths, and, second, the establishment of a sufficient degree of natural immunity at the end of the measures, in order to exclude a second wave. Moreover, the socio-economic costs of the intervention shall be kept at a minimum. The numerically computed optimal control strategy is a single-intervention scenario that goes beyond heuristically motivated interventions and simple "flattening of the curve". Careful analysis of the computed control strategy reveals, however, that the obtained solution is in fact a tightrope walk close to the stability boundary of the system, where socio-economic costs and the risk of a new outbreak must be constantly balanced against one another. The model system is calibrated to reproduce the initial exponential growth phase of the COVID-19 pandemic in Germany.
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Affiliation(s)
- Markus Kantner
- Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, Berlin, 10117 Germany
| | - Thomas Koprucki
- Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, Berlin, 10117 Germany
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