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Andreu-Vilarroig C, Villanueva RJ, González-Parra G. Mathematical modeling for estimating influenza vaccine efficacy: A case study of the Valencian Community, Spain. Infect Dis Model 2024; 9:744-762. [PMID: 38689854 PMCID: PMC11058883 DOI: 10.1016/j.idm.2024.04.006] [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: 02/23/2024] [Revised: 04/02/2024] [Accepted: 04/10/2024] [Indexed: 05/02/2024] Open
Abstract
Vaccine efficacy and its quantification is a crucial concept for the proper design of public health vaccination policies. In this work we proposed a mathematical model to estimate the efficacy of the influenza vaccine in a real-word scenario. In particular, our model is a SEIR-type epidemiological model, which distinguishes vaccinated and unvaccinated populations. Mathematically, its dynamics is governed by a nonlinear system of ordinary differential equations, where the non-linearity arises from the effective contacts between susceptible and infected individuals. Two key aspects of this study is that we use a vaccine distribution over time that is based on real data specific to the elderly people in the Valencian Community and the calibration process takes into account that over one influenza season a specific proportion of the population becomes infected with influenza. To consider the effectiveness of the vaccine, the model incorporates a parameter, the vaccine attenuation factor, which is related with the vaccine efficacy against the influenza virus. With this framework, in order to calibrate the model parameters and to obtain an influenza vaccine efficacy estimation, we considered the 2016-2017 influenza season in the Valencian Community, Spain, using the influenza reported cases of vaccinated and unvaccinated. In order to ensure the model identifiability, we choose to deterministically calibrate the parameters for different scenarios and we find the one with the minimum error in order to determine the vaccine efficacy. The calibration results suggest that the influenza vaccine developed for 2016-2017 influenza season has an efficacy of approximately 76.7%, and that the risk of becoming infected is five times higher for an unvaccinated individual in comparison with a vaccinated one. This estimation partially agrees with some previous studies related to the influenza vaccine. This study presents a new integrated mathematical approach to study the influenza vaccine efficacy and gives further insight into this important public health topic.
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Affiliation(s)
- Carlos Andreu-Vilarroig
- Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, Spain
| | - Rafael J. Villanueva
- Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, Spain
| | - Gilberto González-Parra
- Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, Spain
- Department of Mathematics, New Mexico Tech, Socorro, NM, USA
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Demongeot J, Magal P. Data-driven mathematical modeling approaches for COVID-19: A survey. Phys Life Rev 2024; 50:166-208. [PMID: 39142261 DOI: 10.1016/j.plrev.2024.08.004] [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: 07/15/2024] [Accepted: 08/02/2024] [Indexed: 08/16/2024]
Abstract
In this review, we successively present the methods for phenomenological modeling of the evolution of reported and unreported cases of COVID-19, both in the exponential phase of growth and then in a complete epidemic wave. After the case of an isolated wave, we present the modeling of several successive waves separated by endemic stationary periods. Then, we treat the case of multi-compartmental models without or with age structure. Eventually, we review the literature, based on 260 articles selected in 11 sections, ranging from the medical survey of hospital cases to forecasting the dynamics of new cases in the general population. This review favors the phenomenological approach over the mechanistic approach in the choice of references and provides simulations of the evolution of the number of observed cases of COVID-19 for 10 states (California, China, France, India, Israel, Japan, New York, Peru, Spain and United Kingdom).
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Affiliation(s)
- Jacques Demongeot
- Université Grenoble Alpes, AGEIS EA7407, La Tronche, F-38700, France.
| | - Pierre Magal
- Department of Mathematics, Faculty of Arts and Sciences, Beijing Normal University, Zhuhai, 519087, China; Univ. Bordeaux, IMB, UMR 5251, Talence, F-33400, France; CNRS, IMB, UMR 5251, Talence, F-33400, France
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3
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Webb G, Zhao XE. An Epidemic Model with Infection Age and Vaccination Age Structure. Infect Dis Rep 2024; 16:35-64. [PMID: 38247976 PMCID: PMC10801629 DOI: 10.3390/idr16010004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 11/17/2023] [Revised: 12/27/2023] [Accepted: 01/01/2024] [Indexed: 01/23/2024] Open
Abstract
A model of epidemic dynamics is developed that incorporates continuous variables for infection age and vaccination age. The model analyzes pre-symptomatic and symptomatic periods of an infected individual in terms of infection age. This property is shown to be of major importance in the severity of the epidemic, when the infectious period of an infected individual precedes the symptomatic period. The model also analyzes the efficacy of vaccination in terms of vaccination age. The immunity to infection of vaccinated individuals varies with vaccination age and is also of major significance in the severity of the epidemic. Application of the model to the 2003 SARS epidemic in Taiwan and the COVID-19 epidemic in New York provides insights into the dynamics of these diseases. It is shown that the SARS outbreak was effectively contained due to the complete overlap of infectious and symptomatic periods, allowing for the timely isolation of affected individuals. In contrast, the pre-symptomatic spread of COVID-19 in New York led to a rapid, uncontrolled epidemic. These findings underscore the critical importance of the pre-symptomatic infectious period and the vaccination strategies in influencing the dynamics of an epidemic.
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Affiliation(s)
- Glenn Webb
- Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA
| | - Xinyue Evelyn Zhao
- Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA
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Capistrán MA, Infante JA, Ramos ÁM, Rey JM. Disentangling the role of virus infectiousness and awareness-based human behavior during the early phase of the COVID-19 pandemic in the European Union. APPLIED MATHEMATICAL MODELLING 2023; 122:187-199. [PMID: 37283821 PMCID: PMC10225339 DOI: 10.1016/j.apm.2023.05.027] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/25/2022] [Revised: 04/23/2023] [Accepted: 05/22/2023] [Indexed: 06/08/2023]
Abstract
In this work, we manage to disentangle the role of virus infectiousness and awareness-based human behavior in the COVID-19 pandemic. Using Bayesian inference, we quantify the uncertainty of a state-space model whose propagator is based on an unusual SEIR-type model since it incorporates the effective population fraction as a parameter. Within the Markov Chain Monte Carlo (MCMC) algorithm, Unscented Kalman Filter (UKF) may be used to evaluate the likelihood approximately. UKF is a suitable strategy in many cases, but it is not well-suited to deal with non-negativity restrictions on the state variables. To overcome this difficulty, we modify the UKF, conveniently truncating Gaussian distributions, which allows us to deal with such restrictions. We use official infection notification records to analyze the first 22 weeks of infection spread in each of the 27 countries of the European Union (EU). It is known that such records are the primary source of information to assess the early evolution of the pandemic and, at the same time, usually suffer underreporting and backlogs. Our model explicitly accounts for uncertainty in the dynamic model parameters, the dynamic model adequacy, and the infection observation process. We argue that this modeling paradigm allows us to disentangle the role of the contact rate, the effective population fraction, and the infection observation probability across time and space with an imperfect first principles model. Our findings agree with phylogenetic evidence showing little variability in the contact rate, or virus infectiousness, across EU countries during the early phase of the pandemic, highlighting the advantage of incorporating the effective population fraction into pandemic modeling for heterogeneity in both human behavior and reporting. Finally, to evaluate the consistency of our data assimilation method, we performed a forecast that adequately fits the actual data. Statement of significance Data-driven and model-based epidemiological studies aimed at learning the number of people infected early during a pandemic should explicitly consider the behavior-induced effective population effect. Indeed, the non-isolated, or effective, fraction of the population during the early phase of the pandemic is time-varying, and first-principles modeling with quantified uncertainty is imperative for an adequate analysis across time and space. We argue that, although good inference results may be obtained using the classical SEIR type model, the model posed in this work has allowed us to disentangle the role of virus infectiousness and awareness-based human behavior during the early phase of the COVID-19 pandemic in the European Union from official infection notification records.
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Affiliation(s)
- Marcos A Capistrán
- Centro de Investigación en Matemáticas (CIMAT), Jalisco S/N, Valenciana, Guanajuato 36023, México
| | - Juan-Antonio Infante
- Instituto de Matemática Interdisciplinar and Departamento de Análisis Matemático y Matemática Aplicada, Facultad de CC. Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid, 28040, Spain
| | - Ángel M Ramos
- Instituto de Matemática Interdisciplinar and Departamento de Análisis Matemático y Matemática Aplicada, Facultad de CC. Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid, 28040, Spain
| | - José M Rey
- Instituto de Matemática Interdisciplinar and Departamento de Análisis Matemático y Matemática Aplicada, Facultad de CC. Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid, 28040, Spain
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Moriña D, Fernández-Fontelo A, Cabaña A, Arratia A, Puig P. Estimated Covid-19 burden in Spain: ARCH underreported non-stationary time series. BMC Med Res Methodol 2023; 23:75. [PMID: 36977977 PMCID: PMC10043853 DOI: 10.1186/s12874-023-01894-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2022] [Accepted: 03/17/2023] [Indexed: 03/30/2023] Open
Abstract
BACKGROUND The problem of dealing with misreported data is very common in a wide range of contexts for different reasons. The current situation caused by the Covid-19 worldwide pandemic is a clear example, where the data provided by official sources were not always reliable due to data collection issues and to the high proportion of asymptomatic cases. In this work, a flexible framework is proposed, with the objective of quantifying the severity of misreporting in a time series and reconstructing the most likely evolution of the process. METHODS The performance of Bayesian Synthetic Likelihood to estimate the parameters of a model based on AutoRegressive Conditional Heteroskedastic time series capable of dealing with misreported information and to reconstruct the most likely evolution of the phenomenon is assessed through a comprehensive simulation study and illustrated by reconstructing the weekly Covid-19 incidence in each Spanish Autonomous Community. RESULTS Only around 51% of the Covid-19 cases in the period 2020/02/23-2022/02/27 were reported in Spain, showing relevant differences in the severity of underreporting across the regions. CONCLUSIONS The proposed methodology provides public health decision-makers with a valuable tool in order to improve the assessment of a disease evolution under different scenarios.
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Affiliation(s)
- David Moriña
- Department of Econometrics, Statistics and Applied Economics, Riskcenter-IREA, Universitat de Barcelona (UB), Barcelona, Spain.
| | - Amanda Fernández-Fontelo
- Departament de Matemàtiques, Universitat Autònoma de Barcelona (UAB), Cerdanyola del Vallès, Spain
| | - Alejandra Cabaña
- Departament de Matemàtiques, Universitat Autònoma de Barcelona (UAB), Cerdanyola del Vallès, Spain
| | - Argimiro Arratia
- Department of Computer Science, Universitat Politècnica de Catalunya (UPC), Barcelona, Spain
| | - Pedro Puig
- Departament de Matemàtiques, Universitat Autònoma de Barcelona (UAB), Cerdanyola del Vallès, Spain
- Centre de Recerca Matemàtica (CRM), Barcelona, Spain
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Mathematical Analysis of Epidemic Models with Treatment in Heterogeneous Networks. Bull Math Biol 2023; 85:11. [PMID: 36602636 PMCID: PMC9813916 DOI: 10.1007/s11538-022-01116-1] [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: 05/04/2022] [Accepted: 12/26/2022] [Indexed: 01/06/2023]
Abstract
In this paper, we formulate two different network-based epidemic models to investigate the effect of partly effective treatment on disease dynamics. The first network model represents the individuals with heterogeneous number of contacts in a population as choosing a new partner at each moment, whereas the second one assumes the individuals have fixed or stable neighbors. The basic reproduction number [Formula: see text] is computed for each model, using the next generation matrix method. In particular, the critical treatment rate is defined for the model, above which the disease can be eliminated through the treatment. The final epidemic size relations are derived, and the solvability of these implicit equations is studied. In particular, a unique solution of the implicit equation for the final epidemic size is determined, and by rewriting the implicit equation as a suitable fixed point problem, it is proved that the iteration of the fixed point problem converges to the unique solution. Stochastic simulations and numerical simulations, including in comparison with the model outputs and the joint influence of network topology and treatment on the final epidemic size, are conducted to illustrate the theoretical results.
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HONFO SEWANOUH, TABOE HEMAHOB, KAKAÏ ROMAINGLELE. Modeling COVID-19 dynamics in the sixteen West African countries. SCIENTIFIC AFRICAN 2022; 18:e01408. [PMCID: PMC9621612 DOI: 10.1016/j.sciaf.2022.e01408] [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: 09/19/2020] [Revised: 11/27/2021] [Accepted: 10/20/2022] [Indexed: 11/05/2022] Open
Abstract
The current COVID-19 pandemic has caused several damages to the world, especially in public health sector. This study considered a simple deterministic SIR (Susceptible-Infectious-Recovered) model to characterize and predict future course of the pandemic in the West African countries. We estimated specific characteristics of the disease’s dynamics such as its initial conditions, reproduction numbers, true peak, reported peak with their corresponding times, final epidemic size and time-varying attack ratio. Our findings revealed a relatively low proportion of susceptible individuals in the region and in the different countries (1.2 % across West Africa). The detection rate of the disease was also relatively low (0.9 % for West Africa as a whole) and <2 % for most countries, except for Gambia (12.5 %), Cape-Verde (9.5 %), Mauritania (5.9 %) and Ghana (4.4 %). The reproduction number varied between 1.15 (Burkina-Faso) and 4.45 (Niger) and the peak time of the pandemic was between June and July for most countries. Most generally, the peak time of reported cases came a week (7-8 days) after the true peak time. The model predicted 222,100 actual active cases in the region at peak time while the final epidemic size accounted for 0.6 % of the West African population (2,526,700 individuals). Results obtained showed that the COVID-19 pandemic has not severely affected West Africa as noticed in other regions of the world, but current control measures and standard operating procedures should be maintained over time to ensure trends observed and even accelerate the declining trend of the pandemic.
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Campillo-Funollet E, Wragg H, Van Yperen J, Duong DL, Madzvamuse A. Reformulating the susceptible-infectious-removed model in terms of the number of detected cases: well-posedness of the observational model. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2022; 380:20210306. [PMID: 35965462 PMCID: PMC9376718 DOI: 10.1098/rsta.2021.0306] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/05/2021] [Accepted: 02/23/2022] [Indexed: 06/15/2023]
Abstract
Compartmental models are popular in the mathematics of epidemiology for their simplicity and wide range of applications. Although they are typically solved as initial value problems for a system of ordinary differential equations, the observed data are typically akin to a boundary value-type problem: we observe some of the dependent variables at given times, but we do not know the initial conditions. In this paper, we reformulate the classical susceptible-infectious-recovered system in terms of the number of detected positive infected cases at different times to yield what we term the observational model. We then prove the existence and uniqueness of a solution to the boundary value problem associated with the observational model and present a numerical algorithm to approximate the solution. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.
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Affiliation(s)
- Eduard Campillo-Funollet
- Department of Statistical Methodology and Applications, School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, Kent CT2 7PE, UK
| | - Hayley Wragg
- Department of Engineering Mathematics, School of Computer Science, Electrical and Electronic Engineering and Engineering Maths, University of Bristol, Bristol BS8 1TW, UK
- Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton, East Sussex BN1 9QH, UK
| | - James Van Yperen
- Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton, East Sussex BN1 9QH, UK
| | - Duc-Lam Duong
- Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton, East Sussex BN1 9QH, UK
- School of Engineering Science, LUT University, Lappeenranta 53850, Finland
| | - Anotida Madzvamuse
- Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Brighton, East Sussex BN1 9QH, UK
- Department of Mathematics, University of Johannesburg, Johannesburg, South Africa
- University of British Columbia, Department of Mathematics, Vancouver, Canada
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Prieto K, Chávez–Hernández MV, Romero–Leiton JP. On mobility trends analysis of COVID-19 dissemination in Mexico City. PLoS One 2022; 17:e0263367. [PMID: 35143548 PMCID: PMC8830699 DOI: 10.1371/journal.pone.0263367] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/21/2021] [Accepted: 01/18/2022] [Indexed: 01/04/2023] Open
Abstract
This work presents a tool for forecasting the spread of the new coronavirus in Mexico City, which is based on a mathematical model with a metapopulation structure that uses Bayesian statistics and is inspired by a data-driven approach. The daily mobility of people in Mexico City is mathematically represented by an origin-destination matrix using the open mobility data from Google and the Transportation Mexican Survey. This matrix is incorporated in a compartmental model. We calibrate the model against borough-level incidence data collected between 27 February 2020 and 27 October 2020, while using Bayesian inference to estimate critical epidemiological characteristics associated with the coronavirus spread. Given that working with metapopulation models leads to rather high computational time consumption, and parameter estimation of these models may lead to high memory RAM consumption, we do a clustering analysis that is based on mobility trends to work on these clusters of borough separately instead of taken all of the boroughs together at once. This clustering analysis can be implemented in smaller or larger scales in different parts of the world. In addition, this clustering analysis is divided into the phases that the government of Mexico City has set up to restrict individual movement in the city. We also calculate the reproductive number in Mexico City using the next generation operator method and the inferred model parameters obtaining that this threshold is in the interval (1.2713, 1.3054). Our analysis of mobility trends can be helpful when making public health decisions.
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Affiliation(s)
- Kernel Prieto
- Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico, México
| | - M. Victoria Chávez–Hernández
- Facultad de Ingeniería Mecánica y Eléctrica, Universidad Autónoma de Nuevo León, San Nicolás de los Garza, Mexico, México
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Prieto K. Current forecast of COVID-19 in Mexico: A Bayesian and machine learning approaches. PLoS One 2022; 17:e0259958. [PMID: 35061688 PMCID: PMC8782335 DOI: 10.1371/journal.pone.0259958] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2021] [Accepted: 10/29/2021] [Indexed: 12/24/2022] Open
Abstract
The COVID-19 pandemic has been widely spread and affected millions of people and caused hundreds of deaths worldwide, especially in patients with comorbilities and COVID-19. This manuscript aims to present models to predict, firstly, the number of coronavirus cases and secondly, the hospital care demand and mortality based on COVID-19 patients who have been diagnosed with other diseases. For the first part, I present a projection of the spread of coronavirus in Mexico, which is based on a contact tracing model using Bayesian inference. I investigate the health profile of individuals diagnosed with coronavirus to predict their type of patient care (inpatient or outpatient) and survival. Specifically, I analyze the comorbidity associated with coronavirus using Machine Learning. I have implemented two classifiers: I use the first classifier to predict the type of care procedure that a person diagnosed with coronavirus presenting chronic diseases will obtain (i.e. outpatient or hospitalised), in this way I estimate the hospital care demand; I use the second classifier to predict the survival or mortality of the patient (i.e. survived or deceased). I present two techniques to deal with these kinds of unbalanced datasets related to outpatient/hospitalised and survived/deceased cases (which occur in general for these types of coronavirus datasets) to obtain a better performance for the classification.
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Affiliation(s)
- Kernel Prieto
- Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, México
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11
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New statistical model for misreported data with application to current public health challenges. Sci Rep 2021; 11:23321. [PMID: 34857815 PMCID: PMC8640038 DOI: 10.1038/s41598-021-02620-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/09/2020] [Accepted: 11/18/2021] [Indexed: 01/19/2023] Open
Abstract
The main goal of this work is to present a new model able to deal with potentially misreported continuous time series. The proposed model is able to handle the autocorrelation structure in continuous time series data, which might be partially or totally underreported or overreported. Its performance is illustrated through a comprehensive simulation study considering several autocorrelation structures and three real data applications on human papillomavirus incidence in Girona (Catalonia, Spain) and Covid-19 incidence in two regions with very different circumstances: the early days of the epidemic in the Chinese region of Heilongjiang and the most current data from Catalonia.
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Tiomela SA, Macías-Díaz JE, Mvogo A. Computer simulation of the dynamics of a spatial susceptible-infected-recovered epidemic model with time delays in transmission and treatment. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2021; 212:106469. [PMID: 34715516 DOI: 10.1016/j.cmpb.2021.106469] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2021] [Accepted: 10/08/2021] [Indexed: 06/13/2023]
Abstract
BACKGROUND AND OBJECTIVE In this work, we analyze the spatial-temporal dynamics of a susceptible-infected-recovered (SIR) epidemic model with time delays. To better describe the dynamical behavior of the model, we take into account the cumulative effects of diffusion in the population dynamics, and the time delays in both the Holling type II treatment and the disease transmission process, respectively. METHODS We perform linear stability analyses on the disease-free and endemic equilibria. We provide the expression of the basic reproduction number and set conditions on the backward bifurcation using Castillo's theorem. The values of the critical time transmission, the treatment delays and the relationship between them are established. RESULTS We show that the treatment rate decreases the basic reproduction number while the transmission rate significantly affects the bifurcation process in the system. The transmission and treatment time-delays are found to be inversely proportional to the susceptible and infected diffusion rates. The analytical results are numerically tested. The results show that the treatment rate significantly reduces the density of infected population and ensures the transition from the unstable to the stable domain. Moreover, the system is more sensible to the treatment in the stable domain. CONCLUSIONS The density of infected population increases with respect to the infected and susceptible diffusion rates. Both effects of treatment and transmission delays significantly affect the behavior of the system. The transmission time-delay at the critical point ensures the transition from the stable (low density) to the unstable (high density) domain.
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Affiliation(s)
- Sedrique A Tiomela
- Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, Yaoundé P.O. Box 812, Cameroon.
| | - J E Macías-Díaz
- Department of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, Tallinn 10120, Estonia; Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20131, Mexico.
| | - Alain Mvogo
- Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, Yaoundé P.O. Box 812, Cameroon.
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Fotsa-Mbogne DJ, Tchoumi SY, Kouakep-Tchaptchie Y, Kamla VC, Kamgang JC, Houpa-Danga DE, Bowong-Tsakou S, Bekolle D. Estimation and optimal control of the multiscale dynamics of Covid-19: a case study from Cameroon. NONLINEAR DYNAMICS 2021; 106:2703-2738. [PMID: 34697521 PMCID: PMC8528969 DOI: 10.1007/s11071-021-06920-3] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2021] [Accepted: 09/18/2021] [Indexed: 05/31/2023]
Abstract
This work aims at a better understanding and the optimal control of the spread of the new severe acute respiratory corona virus 2 (SARS-CoV-2). A multi-scale model giving insights on the virus population dynamics, the transmission process and the infection mechanism is proposed first. Indeed, there are human to human virus transmission, human to environment virus transmission, environment to human virus transmission and self-infection by susceptible individuals. The global stability of the disease-free equilibrium is shown when a given threshold T 0 is less or equal to 1 and the basic reproduction number R 0 is calculated. A convergence index T 1 is also defined in order to estimate the speed at which the disease extincts and an upper bound to the time of infectious extinction is given. The existence of the endemic equilibrium is conditional and its description is provided. Using Partial Rank Correlation Coefficient with a three levels fractional experimental design, the sensitivity of R 0 , T 0 and T 1 to control parameters is evaluated. Following this study, the most significant parameter is the probability of wearing mask followed by the probability of mobility and the disinfection rate. According to a functional cost taking into account economic impacts of SARS-CoV-2, optimal fighting strategies are determined and discussed. The study is applied to real and available data from Cameroon with a model fitting. After several simulations, social distancing and the disinfection frequency appear as the main elements of the optimal control strategy against SARS-CoV-2.
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Affiliation(s)
- David Jaurès Fotsa-Mbogne
- Department of Mathematics and Computer Science, ENSAI, The University of Ngaoundere, P.O. Box 455, Ngaoundere, Cameroon
| | - Stéphane Yanick Tchoumi
- Department of Mathematics and Computer Science, ENSAI, The University of Ngaoundere, P.O. Box 455, Ngaoundere, Cameroon
| | - Yannick Kouakep-Tchaptchie
- Department of Fundamental Science and Engineering, EGCIM, The University of Ngaoundere, P.O. Box 454, Ngaoundere, Cameroon
| | - Vivient Corneille Kamla
- Department of Mathematics and Computer Science, ENSAI, The University of Ngaoundere, P.O. Box 455, Ngaoundere, Cameroon
| | - Jean-Claude Kamgang
- Department of Mathematics and Computer Science, ENSAI, The University of Ngaoundere, P.O. Box 455, Ngaoundere, Cameroon
| | - Duplex Elvis Houpa-Danga
- Department of Mathematics and Computer Science, FS, The University of Ngaoundere, P.O. Box 454, Ngaoundere, Cameroon
| | - Samuel Bowong-Tsakou
- Department of Mathematics and Computer Science, FS, The University of Douala, P.O. Box 24157, Douala, Cameroon
| | - David Bekolle
- Department of Mathematics and Computer Science, FS, The University of Ngaoundere, P.O. Box 454, Ngaoundere, Cameroon
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Iqbal Z, Macías-Díaz JE, Ahmed N, Aziz-Ur Rehman M, Raza A, Rafiq M. A SEIR model with memory effects for the propagation of Ebola-like infections and its dynamically consistent approximation. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2021; 209:106322. [PMID: 34390936 DOI: 10.1016/j.cmpb.2021.106322] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/14/2021] [Accepted: 07/24/2021] [Indexed: 06/13/2023]
Abstract
BACKGROUND AND OBJECTIVE We present and analyze a nonstandard numerical method to solve an epidemic model with memory that describes the propagation of Ebola-type diseases. The epidemiological system contemplates the presence of sub-populations of susceptible, exposed, infected and recovered individuals, along with nonlinear interactions between the members of those sub-populations. The system possesses disease-free and endemic equilibrium points, whose stability is studied rigorously. METHODS To solve the epidemic model with memory, a nonstandard approach based on Grünwald-Letnikov differences is used to discretize the problem. The discretization is conveniently carried out in order to produce a fully explicit and non-singular scheme. The discrete problem is thus well defined for any set of non-negative initial conditions. RESULTS The existence and uniqueness of the solutions of the discrete problem for non-negative initial data is thoroughly proved. Moreover, the positivity and the boundedness of the approximations is also theoretically elucidated. Some simulations confirm the validity of these theoretical results. Moreover, the simulations prove that the computational model is capable of preserving the equilibria of the system (both the disease-free and the endemic equilibria) as well as the stability of those points. CONCLUSIONS Both theoretical and numerical results establish that the computational method proposed in this work is capable of preserving distinctive features of an epidemiological model with memory for the propagation of Ebola-type diseases. Among the main characteristics of the numerical integrator, the existence and the uniqueness of solutions, the preservation of both positivity and boundedness, the preservation of the equilibrium points and their stabilities as well as the easiness to implement it computationally are the most important features of the approach proposed in this manuscript.
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Affiliation(s)
- Zafar Iqbal
- Department of Mathematics, University of Management and Technology, Lahore, Pakistan; Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan.
| | - J E Macías-Díaz
- Department of Mathematics, School of Digital Technologies, Tallinn University, Estonia; Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes, 20131, Mexico.
| | - Nauman Ahmed
- Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan.
| | - M Aziz-Ur Rehman
- Department of Mathematics, University of Management and Technology, Lahore, Pakistan.
| | - Ali Raza
- Department of Mathematics, National College of Business, Administration and Economics Lahore, Pakistan.
| | - Muhammad Rafiq
- Department of Mathematics, Faculty of Sciences, University of Central Punjab, Lahore, Pakistan.
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15
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Webb G. A COVID-19 Epidemic Model Predicting the Effectiveness of Vaccination in the US. Infect Dis Rep 2021; 13:654-667. [PMID: 34449651 PMCID: PMC8395902 DOI: 10.3390/idr13030062] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 06/26/2021] [Revised: 07/21/2021] [Accepted: 07/22/2021] [Indexed: 12/17/2022] Open
Abstract
A model of a COVID-19 epidemic is used to predict the effectiveness of vaccination in the US. The model incorporates key features of COVID-19 epidemics: asymptomatic and symptomatic infectiousness, reported and unreported cases data, and social measures implemented to decrease infection transmission. The model analyzes the effectiveness of vaccination in terms of vaccination efficiency, vaccination scheduling, and relaxation of social measures that decrease disease transmission. The model demonstrates that the subsiding of the epidemic as vaccination is implemented depends critically on the scale of relaxation of social measures that reduce disease transmission.
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Affiliation(s)
- Glenn Webb
- Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA
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16
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Zawadzki RS, Gong CL, Cho SK, Schnitzer JE, Zawadzki NK, Hay JW, Drabo EF. Where Do We Go From Here? A Framework for Using Susceptible-Infectious-Recovered Models for Policy Making in Emerging Infectious Diseases. VALUE IN HEALTH : THE JOURNAL OF THE INTERNATIONAL SOCIETY FOR PHARMACOECONOMICS AND OUTCOMES RESEARCH 2021; 24:917-924. [PMID: 34243834 PMCID: PMC8110035 DOI: 10.1016/j.jval.2021.03.005] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/22/2020] [Revised: 03/03/2021] [Accepted: 03/07/2021] [Indexed: 06/13/2023]
Abstract
OBJECTIVES Throughout the coronavirus disease 2019 pandemic, susceptible-infectious-recovered (SIR) modeling has been the preeminent modeling method to inform policy making worldwide. Nevertheless, the usefulness of such models has been subject to controversy. An evolution in the epidemiological modeling field is urgently needed, beginning with an agreed-upon set of modeling standards for policy recommendations. The objective of this article is to propose a set of modeling standards to support policy decision making. METHODS We identify and describe 5 broad standards: transparency, heterogeneity, calibration and validation, cost-benefit analysis, and model obsolescence and recalibration. We give methodological recommendations and provide examples in the literature that employ these standards well. We also develop and demonstrate a modeling practices checklist using existing coronavirus disease 2019 literature that can be employed by readers, authors, and reviewers to evaluate and compare policy modeling literature along our formulated standards. RESULTS We graded 16 articles using our checklist. On average, the articles met 6.81 of our 19 categories (36.7%). No articles contained any cost-benefit analyses and few were adequately transparent. CONCLUSIONS There is significant room for improvement in modeling pandemic policy. Issues often arise from a lack of transparency, poor modeling assumptions, lack of a system-wide perspective in modeling, and lack of flexibility in the academic system to rapidly iterate modeling as new information becomes available. In anticipation of future challenges, we encourage the modeling community at large to contribute toward the refinement and consensus of a shared set of standards for infectious disease policy modeling.
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Affiliation(s)
- Roy S Zawadzki
- Department of Statistics, Donald Bren School of Information and Computer Sciences, University of California, Irvine, CA, USA
| | - Cynthia L Gong
- Fetal and Neonatal Institute, Division of Neonatology, Department of Pediatrics, Children's Hospital Los Angeles, Keck School of Medicine, University of Southern California, Los Angeles, CA, USA.
| | - Sang K Cho
- College of Pharmacy, University of Houston, Houston, TX, USA
| | - Jan E Schnitzer
- Proteogenomics Research Institute for Systems Medicine (PRISM), San Diego, CA, USA
| | - Nadine K Zawadzki
- Schaeffer Center for Health Policy & Economics, School of Pharmacy, University of Southern California, Los Angeles, CA, USA
| | - Joel W Hay
- Schaeffer Center for Health Policy & Economics, School of Pharmacy, University of Southern California, Los Angeles, CA, USA
| | - Emmanuel F Drabo
- Department of Health Policy and Management, Bloomberg School of Public Health, Johns Hopkins University, Baltimore, MD, USA
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17
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Sallahi N, Park H, El Mellouhi F, Rachdi M, Ouassou I, Belhaouari S, Arredouani A, Bensmail H. Using Unstated Cases to Correct for COVID-19 Pandemic Outbreak and Its Impact on Easing the Intervention for Qatar. BIOLOGY 2021; 10:biology10060463. [PMID: 34073810 PMCID: PMC8225146 DOI: 10.3390/biology10060463] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 04/07/2021] [Revised: 05/16/2021] [Accepted: 05/19/2021] [Indexed: 12/24/2022]
Abstract
Simple Summary A modified SIR model was applied to provide COVID-19 pandemic analysis and predictions for Gulf Cooperation Council countries, as well as representative countries in Europe and New York City. We estimated reported, infected, and unreported cases from cumulative reported cases and simulated data. We also estimated the basic reproduction rates at different phases of the pandemic. Outputs show that the modified SIR model fits very well with the outcome of the COVID-19 pandemic for the studied countries and could be generalized to other countries. The model prediction emphasizes the value of significant interventions in public health in regulating the epidemic taking into account that a constant fraction of the infected cases remain unreported during the pandemic. We report and analyze the effectiveness of preventive/intervention measures applied to the overall community to curb the severity of the pandemic. Our model could be used to support public health authorities with respect to post-outbreak reopening decisions, highlighting effective measures that need to be maintained, eased, or implemented to support safe reopening strategies in the GCC countries. Abstract Epidemiological Modeling supports the evaluation of various disease management activities. The value of epidemiological models lies in their ability to study various scenarios and to provide governments with a priori knowledge of the consequence of disease incursions and the impact of preventive strategies. A prevalent method of modeling the spread of pandemics is to categorize individuals in the population as belonging to one of several distinct compartments, which represents their health status with regard to the pandemic. In this work, a modified SIR epidemic model is proposed and analyzed with respect to the identification of its parameters and initial values based on stated or recorded case data from public health sources to estimate the unreported cases and the effectiveness of public health policies such as social distancing in slowing the spread of the epidemic. The analysis aims to highlight the importance of unreported cases for correcting the underestimated basic reproduction number. In many epidemic outbreaks, the number of reported infections is likely much lower than the actual number of infections which can be calculated from the model’s parameters derived from reported case data. The analysis is applied to the COVID-19 pandemic for several countries in the Gulf region and Europe.
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Affiliation(s)
- Narjiss Sallahi
- National Institute of Posts and Telecommunications (INPT), Rabat 210024, Morocco;
| | - Heesoo Park
- Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, Doha P.O. Box 34110, Qatar; (H.P.); (F.E.M.)
| | - Fedwa El Mellouhi
- Qatar Environment and Energy Research Institute, Hamad Bin Khalifa University, Doha P.O. Box 34110, Qatar; (H.P.); (F.E.M.)
| | - Mustapha Rachdi
- Data Sciences Project, University of Grenoble, 38400 Grenoble, France;
| | - Idir Ouassou
- University Qaddi Ayyad, Marrakech 40000, Morocco;
| | - Samir Belhaouari
- ICT Department, Hamad Bin Khalifa University, Doha P.O. Box 34110, Qatar;
| | - Abdelilah Arredouani
- Diabetes Department, Qatar Biomedical Research Institute, Hamad Bin Khalifa University, Doha P.O. Box 5825, Qatar;
| | - Halima Bensmail
- Data Analytics Department, Qatar Computing Research Institute, Hamad Bin Khalifa University, Doha P.O. Box 5825, Qatar
- Correspondence: ; Tel.: +974-5527-8824
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18
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Dynamics of epidemic spreading on connected graphs. J Math Biol 2021; 82:52. [PMID: 33864137 PMCID: PMC8051836 DOI: 10.1007/s00285-021-01602-5] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2020] [Revised: 03/03/2021] [Accepted: 03/26/2021] [Indexed: 01/21/2023]
Abstract
We propose a new model that describes the dynamics of epidemic spreading on connected graphs. Our model consists in a PDE-ODE system where at each vertex of the graph we have a standard SIR model and connections between vertices are given by heat equations on the edges supplemented with Robin like boundary conditions at the vertices modeling exchanges between incident edges and the associated vertex. We describe the main properties of the system, and also derive the final total population of infected individuals. We present a semi-implicit in time numerical scheme based on finite differences in space which preserves the main properties of the continuous model such as the uniqueness and positivity of solutions and the conservation of the total population. We also illustrate our results with a collection of numerical simulations for a selection of connected graphs.
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19
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Waterlow NR, Flasche S, Minter A, Eggo RM. Competition between RSV and influenza: Limits of modelling inference from surveillance data. Epidemics 2021; 35:100460. [PMID: 33838587 PMCID: PMC8193815 DOI: 10.1016/j.epidem.2021.100460] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/27/2020] [Revised: 03/24/2021] [Accepted: 03/24/2021] [Indexed: 10/28/2022] Open
Abstract
Respiratory Syncytial Virus (RSV) and Influenza cause a large burden of disease. Evidence of their interaction via temporary cross-protection implies that prevention of one could inadvertently lead to an increase in the burden of the other. However, evidence for the public health impact of such interaction is sparse and largely derives from ecological analyses of peak shifts in surveillance data. To test the robustness of estimates of interaction parameters between RSV and Influenza from surveillance data we conducted a simulation and back-inference study. We developed a two-pathogen interaction model, parameterised to simulate RSV and Influenza epidemiology in the UK. Using the infection model in combination with a surveillance-like stochastic observation process we generated a range of possible RSV and Influenza trajectories and then used Markov Chain Monte Carlo (MCMC) methods to back-infer parameters including those describing competition. We find that in most scenarios both the strength and duration of RSV and Influenza interaction could be estimated from the simulated surveillance data reasonably well. However, the robustness of inference declined towards the extremes of the plausible parameter ranges, with misleading results. It was for instance not possible to tell the difference between low/moderate interaction and no interaction. In conclusion, our results illustrate that in a plausible parameter range, the strength of RSV and Influenza interaction can be estimated from a single season of high-quality surveillance data but also highlights the importance to test parameter identifiability a priori in such situations.
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Affiliation(s)
- Naomi R Waterlow
- Department of Infectious Disease Epidemiology, London School of Hygiene & Tropical Medicine, UK.
| | - Stefan Flasche
- Department of Infectious Disease Epidemiology, London School of Hygiene & Tropical Medicine, UK
| | - Amanda Minter
- Department of Infectious Disease Epidemiology, London School of Hygiene & Tropical Medicine, UK
| | - Rosalind M Eggo
- Department of Infectious Disease Epidemiology, London School of Hygiene & Tropical Medicine, UK
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20
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Efimov D, Ushirobira R. On an interval prediction of COVID-19 development based on a SEIR epidemic model. ANNUAL REVIEWS IN CONTROL 2021; 51:477-487. [PMID: 33623479 PMCID: PMC7891093 DOI: 10.1016/j.arcontrol.2021.01.006] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2020] [Revised: 01/08/2021] [Accepted: 01/26/2021] [Indexed: 05/13/2023]
Abstract
In this paper, a new version of the well-known epidemic mathematical SEIR model is used to analyze the pandemic course of COVID-19 in eight different countries. One of the proposed model's improvements is to reflect the societal feedback on the disease and confinement features. The SEIR model parameters are allowed to be time-varying, and the ranges of their values are identified by using publicly available data for France, Italy, Spain, Germany, Brazil, Russia, New York State (US), and China. The identified model is then applied to predict the SARS-CoV-2 virus propagation under various conditions of confinement. For this purpose, an interval predictor is designed, allowing variations and uncertainties in the model parameters to be taken into account. The code and the utilized data are available on Github.
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Affiliation(s)
- Denis Efimov
- Inria, Univ. Lille, CNRS, UMR 9189 - CRIStAL, F-59000 Lille, France
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21
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Korolev I. Identification and estimation of the SEIRD epidemic model for COVID-19. JOURNAL OF ECONOMETRICS 2021; 220:63-85. [PMID: 32836680 PMCID: PMC7392128 DOI: 10.1016/j.jeconom.2020.07.038] [Citation(s) in RCA: 52] [Impact Index Per Article: 17.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/21/2020] [Revised: 07/12/2020] [Accepted: 07/17/2020] [Indexed: 05/05/2023]
Abstract
This paper studies the SEIRD epidemic model for COVID-19. First, I show that the model is poorly identified from the observed number of deaths and confirmed cases. There are many sets of parameters that are observationally equivalent in the short run but lead to markedly different long run forecasts. Second, I show that the basic reproduction numberR 0 can be identified from the data, conditional on epidemiologic parameters, and propose several nonlinear SUR approaches to estimateR 0 . I examine the performance of these methods using Monte Carlo studies and demonstrate that they yield fairly accurate estimates ofR 0 . Next, I apply these methods to estimateR 0 for the US, California, and Japan, and document heterogeneity in the value ofR 0 across regions. My estimation approach accounts for possible underreporting of the number of cases. I demonstrate that if one fails to take underreporting into account and estimatesR 0 from the reported cases data, the resulting estimate ofR 0 may be biased downward and the resulting forecasts may exaggerate the long run number of deaths. Finally, I discuss how auxiliary information from random tests can be used to calibrate the initial parameters of the model and narrow down the range of possible forecasts of the future number of deaths.
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Affiliation(s)
- Ivan Korolev
- Department of Economics, Binghamton University, 4400 Vestal Parkway East, PO Box 6000, Binghamton, NY 13902-6000, USA
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22
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Hortaçsu A, Liu J, Schwieg T. Estimating the fraction of unreported infections in epidemics with a known epicenter: An application to COVID-19. JOURNAL OF ECONOMETRICS 2021; 220:106-129. [PMID: 32921876 PMCID: PMC7476454 DOI: 10.1016/j.jeconom.2020.07.047] [Citation(s) in RCA: 28] [Impact Index Per Article: 9.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/02/2020] [Revised: 07/04/2020] [Accepted: 07/06/2020] [Indexed: 05/20/2023]
Abstract
We develop an analytically tractable method to estimate the fraction of unreported infections in epidemics with a known epicenter and estimate the number of unreported COVID-19 infections in the U.S. during the first half of March 2020. Our method utilizes the covariation in initial reported infections across U.S. regions and the number of travelers to these regions from the epicenter, along with the results of an early randomized testing study in Iceland. Using our estimates of the number of unreported infections, which are substantially larger than the number of reported infections, we also provide estimates for the infection fatality rate using data on reported COVID-19 fatalities from U.S. counties.
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Affiliation(s)
- Ali Hortaçsu
- Kenneth C. Griffin Department of Economics, University of Chicago, United States of America
- NBER, United States of America
| | - Jiarui Liu
- Kenneth C. Griffin Department of Economics, University of Chicago, United States of America
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23
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Wacker B, Schlüter J. Time-continuous and time-discrete SIR models revisited: theory and applications. ADVANCES IN DIFFERENCE EQUATIONS 2020; 2020:556. [PMID: 33042201 PMCID: PMC7538854 DOI: 10.1186/s13662-020-02995-1] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 05/12/2020] [Accepted: 09/22/2020] [Indexed: 06/11/2023]
Abstract
Since Kermack and McKendrick have introduced their famous epidemiological SIR model in 1927, mathematical epidemiology has grown as an interdisciplinary research discipline including knowledge from biology, computer science, or mathematics. Due to current threatening epidemics such as COVID-19, this interest is continuously rising. As our main goal, we establish an implicit time-discrete SIR (susceptible people-infectious people-recovered people) model. For this purpose, we first introduce its continuous variant with time-varying transmission and recovery rates and, as our first contribution, discuss thoroughly its properties. With respect to these results, we develop different possible time-discrete SIR models, we derive our implicit time-discrete SIR model in contrast to many other works which mainly investigate explicit time-discrete schemes and, as our main contribution, show unique solvability and further desirable properties compared to its continuous version. We thoroughly show that many of the desired properties of the time-continuous case are still valid in the time-discrete implicit case. Especially, we prove an upper error bound for our time-discrete implicit numerical scheme. Finally, we apply our proposed time-discrete SIR model to currently available data regarding the spread of COVID-19 in Germany and Iran.
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Affiliation(s)
- Benjamin Wacker
- Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany
| | - Jan Schlüter
- Next Generation Mobility Group, Department of Dynamics of Complex Fluids, Max-Planck-Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany
- Institute for Dynamics of Complex Fluids, Faculty of Physics, Georg-August-University of Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany
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24
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On Confinement and Quarantine Concerns on an SEIAR Epidemic Model with Simulated Parameterizations for the COVID-19 Pandemic. Symmetry (Basel) 2020. [DOI: 10.3390/sym12101646] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022] Open
Abstract
This paper firstly studies an SIR (susceptible-infectious-recovered) epidemic model without demography and with no disease mortality under both total and under partial quarantine of the susceptible subpopulation or of both the susceptible and the infectious ones in order to satisfy the hospital availability requirements on bed disposal and other necessary treatment means for the seriously infectious subpopulations. The seriously infectious individuals are assumed to be a part of the total infectious being described by a time-varying proportional function. A time-varying upper-bound of those seriously infected individuals has to be satisfied as objective by either a total confinement or partial quarantine intervention of the susceptible subpopulation. Afterwards, a new extended SEIR (susceptible-exposed-infectious-recovered) epidemic model, which is referred to as an SEIAR (susceptible-exposed-symptomatic infectious-asymptomatic infectious-recovered) epidemic model with demography and disease mortality is given and focused on so as to extend the above developed ideas on the SIR model. A proportionally gain in the model parameterization is assumed to distribute the transition from the exposed to the infectious into the two infectious individuals (namely, symptomatic and asymptomatic individuals). Such a model is evaluated under total or partial quarantines of all or of some of the subpopulations which have the effect of decreasing the number of contagions. Simulated numerical examples are also discussed related to model parameterizations of usefulness related to the current COVID-19 pandemic outbreaks.
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25
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Bracher J, Held L. A marginal moment matching approach for fitting endemic-epidemic models to underreported disease surveillance counts. Biometrics 2020; 77:1202-1214. [PMID: 32920842 DOI: 10.1111/biom.13371] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/15/2019] [Accepted: 09/01/2020] [Indexed: 11/30/2022]
Abstract
Count data are often subject to underreporting, especially in infectious disease surveillance. We propose an approximate maximum likelihood method to fit count time series models from the endemic-epidemic class to underreported data. The approach is based on marginal moment matching where underreported processes are approximated through completely observed processes from the same class. Moreover, the form of the bias when underreporting is ignored or taken into account via multiplication factors is analyzed. Notably, we show that this leads to a downward bias in model-based estimates of the effective reproductive number. A marginal moment matching approach can also be used to account for reporting intervals which are longer than the mean serial interval of a disease. The good performance of the proposed methodology is demonstrated in simulation studies. An extension to time-varying parameters and reporting probabilities is discussed and applied in a case study on weekly rotavirus gastroenteritis counts in Berlin, Germany.
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Affiliation(s)
- Johannes Bracher
- Epidemiology, Biostatistics and Prevention Institute, University of Zurich, Zurich, Switzerland
| | - Leonhard Held
- Epidemiology, Biostatistics and Prevention Institute, University of Zurich, Zurich, Switzerland
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26
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Cotta RM, Naveira-Cotta CP, Magal P. Mathematical Parameters of the COVID-19 Epidemic in Brazil and Evaluation of the Impact of Different Public Health Measures. BIOLOGY 2020; 9:E220. [PMID: 32806613 PMCID: PMC7464380 DOI: 10.3390/biology9080220] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/06/2020] [Revised: 08/04/2020] [Accepted: 08/04/2020] [Indexed: 01/08/2023]
Abstract
A SIRU-type epidemic model is employed for the prediction of the COVID-19 epidemy evolution in Brazil, and analyze the influence of public health measures on simulating the control of this infectious disease. The proposed model allows for a time variable functional form of both the transmission rate and the fraction of asymptomatic infectious individuals that become reported symptomatic individuals, to reflect public health interventions, towards the epidemy control. An exponential analytical behavior for the accumulated reported cases evolution is assumed at the onset of the epidemy, for explicitly estimating initial conditions, while a Bayesian inference approach is adopted for the estimation of parameters by employing the direct problem model with the data from the first phase of the epidemy evolution, represented by the time series for the reported cases of infected individuals. The evolution of the COVID-19 epidemy in China is considered for validation purposes, by taking the first part of the dataset of accumulated reported infectious individuals to estimate the related parameters, and retaining the rest of the evolution data for direct comparison with the predicted results. Then, the available data on reported cases in Brazil from 15 February until 29 March, is used for estimating parameters and then predicting the first phase of the epidemy evolution from these initial conditions. The data for the reported cases in Brazil from 30 March until 23 April are reserved for validation of the model. Then, public health interventions are simulated, aimed at evaluating the effects on the disease spreading, by acting on both the transmission rate and the fraction of the total number of the symptomatic infectious individuals, considering time variable exponential behaviors for these two parameters. This first constructed model provides fairly accurate predictions up to day 65 below 5% relative deviation, when the data starts detaching from the theoretical curve. From the simulated public health intervention measures through five different scenarios, it was observed that a combination of careful control of the social distancing relaxation and improved sanitary habits, together with more intensive testing for isolation of symptomatic cases, is essential to achieve the overall control of the disease and avoid a second more strict social distancing intervention. Finally, the full dataset available by the completion of the present work is employed in redefining the model to yield updated epidemy evolution estimates.
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Affiliation(s)
- Renato M Cotta
- General Directorate of Nuclear and Technological Development, DGDNTM, Brazilian Navy, Ilha das Cobras, Centro, Rio de Janeiro, RJ CEP 20091-000, Brazil
- Laboratory of Nano & Microfluidics and Microsystems, LabMEMS, Mechanical Engineering Department, POLI & COPPE, UFRJ, Federal University of Rio de Janeiro, Cidade Universitária, Rio de Janeiro, RJ CEP 21945-970, Brazil
| | - Carolina P Naveira-Cotta
- Laboratory of Nano & Microfluidics and Microsystems, LabMEMS, Mechanical Engineering Department, POLI & COPPE, UFRJ, Federal University of Rio de Janeiro, Cidade Universitária, Rio de Janeiro, RJ CEP 21945-970, Brazil
| | - Pierre Magal
- Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351, COURS de la Libération, 33400 Talence, France
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27
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Jing SL, Huo HF, Xiang H. Modeling the Effects of Meteorological Factors and Unreported Cases on Seasonal Influenza Outbreaks in Gansu Province, China. Bull Math Biol 2020; 82:73. [PMID: 32533498 DOI: 10.1007/s11538-020-00747-6] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/13/2019] [Accepted: 05/14/2020] [Indexed: 10/24/2022]
Abstract
Influenza usually breaks out seasonally in temperate regions, especially in winter, infection rates and mortality rates of influenza increase significantly, which means that dry air and cold temperatures accelerate the spread of influenza viruses. However, the meteorological factors that lead to seasonal influenza outbreaks and how these meteorological factors play a decisive role in influenza transmission remain unclear. During the epidemic of infectious diseases, the neglect of unreported cases leads to an underestimation of infection rates and basic reproduction number. In this paper, we propose a new non-autonomous periodic differential equation model with meteorological factors including unreported cases. First, the basic reproduction number is obtained and the global asymptotic stability of the disease-free periodic solution is proved. Furthermore, the existence of periodic solutions and the uniformly persistence of the model are demonstrated. Second, the best-fit parameter values in our model are identified by the MCMC algorithm on the basis of the influenza data in Gansu province, China. We also estimate that the basic reproduction number is 1.2288 (95% CI:(1.2287, 1.2289)). Then, to determine the key parameters of the model, uncertainty and sensitivity analysis are explored. Finally, our results show that influenza is more likely to spread in low temperature, low humidity and low precipitation environments. Temperature is a more important factor than relative humidity and precipitation during the influenza epidemic. In addition, our results also show that there are far more unreported cases than reported cases.
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Affiliation(s)
- Shuang-Lin Jing
- Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, 730050, Gansu, People's Republic of China
| | - Hai-Feng Huo
- Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, 730050, Gansu, People's Republic of China.
| | - Hong Xiang
- Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, 730050, Gansu, People's Republic of China
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Liu ZH, Magal P, Seydi O, Webb G. Predicting the cumulative number of cases for the COVID-19 epidemic in China from early data. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2020; 17:3040-3051. [PMID: 32987515 DOI: 10.1101/2020.03.11.20034314] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
We model the COVID-19 coronavirus epidemic in China. We use early reported case data to predict the cumulative number of reported cases to a final size. The key features of our model are the timing of implementation of major public policies restricting social movement, the identification and isolation of unreported cases, and the impact of asymptomatic infectious cases.
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Affiliation(s)
- Zhi Hua Liu
- School of Mathematical Sciences, Beijing Normal University. Beijing 100875, China
| | - Pierre Magal
- Université de Bordeaux, IMB, UMR 5251, F-33400 Talence, France
- CNRS, IMB, UMR 5251, F-33400 Talence, France
| | - Ousmane Seydi
- Département Tronc Commun, École Polytechnique de Thiès, Sénégal
| | - Glenn Webb
- Mathematics Department, Vanderbilt University, Nashville, TN, USA
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Liu Z, Magal P, Seydi O, Webb G. Understanding Unreported Cases in the COVID-19 Epidemic Outbreak in Wuhan, China, and the Importance of Major Public Health Interventions. BIOLOGY 2020; 9:biology9030050. [PMID: 32182724 PMCID: PMC7150940 DOI: 10.3390/biology9030050] [Citation(s) in RCA: 111] [Impact Index Per Article: 27.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/04/2020] [Revised: 02/23/2020] [Accepted: 02/28/2020] [Indexed: 01/10/2023]
Abstract
We develop a mathematical model to provide epidemic predictions for the COVID-19 epidemic in Wuhan, China. We use reported case data up to 31 January 2020 from the Chinese Center for Disease Control and Prevention and the Wuhan Municipal Health Commission to parameterize the model. From the parameterized model, we identify the number of unreported cases. We then use the model to project the epidemic forward with varying levels of public health interventions. The model predictions emphasize the importance of major public health interventions in controlling COVID-19 epidemics.
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Affiliation(s)
- Zhihua Liu
- School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China;
| | - Pierre Magal
- Université de Bordeaux, IMB, UMR 5251, F-33400 Talence, France
- CNRS, IMB, UMR 5251, F-33400 Talence, France
- Correspondence:
| | - Ousmane Seydi
- Département Tronc Commun, École Polytechnique de Thiés, Thies 21001, Senegal;
| | - Glenn Webb
- Mathematics Department, Vanderbilt University, Nashville, TN 37212, USA;
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Ducrot A, Magal P, Nguyen T, Webb GF. Identifying the number of unreported cases in SIR epidemic models. MATHEMATICAL MEDICINE AND BIOLOGY-A JOURNAL OF THE IMA 2019; 37:243-261. [DOI: 10.1093/imammb/dqz013] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2018] [Revised: 05/13/2019] [Accepted: 05/21/2019] [Indexed: 11/13/2022]
Abstract
Abstract
An SIR epidemic model is analysed with respect to the identification of its parameters and initial values, based upon reported case data from public health sources. The objective of the analysis is to understand the relationship of unreported cases to reported cases. In many epidemic diseases the reported cases are a small fraction of the unreported cases. This fraction can be estimated by the identification of parameters for the model from reported case data. The analysis is applied to the Hong Kong seasonal influenza epidemic in New York City in 1968–1969.
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Affiliation(s)
- A Ducrot
- Normandie University, UNIHAVRE, LMAH, FR-CNRS-3335, ISCN, Le Havre, France
| | - P Magal
- University of Bordeaux, CNRS, Bordeaux INP, IMB, UMR, Talence, France
| | - T Nguyen
- University of Bordeaux, CNRS, Bordeaux INP, IMB, UMR, Talence, France
| | - G F Webb
- Mathematics Department, Vanderbilt University, Nashville, TN, USA
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On a SIR Model in a Patchy Environment Under Constant and Feedback Decentralized Controls with Asymmetric Parameterizations. Symmetry (Basel) 2019. [DOI: 10.3390/sym11030430] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
: This paper presents a formal description and analysis of an SIR (involving susceptible- infectious-recovered subpopulations) epidemic model in a patchy environment with vaccination controls being constant and proportional to the susceptible subpopulations. The patchy environment is due to the fact that there is a partial interchange of all the subpopulations considered in the model between the various patches what is modelled through the so-called travel matrices. It is assumed that the vaccination controls are administered at each community health centre of a particular patch while either the total information or a partial information of the total subpopulations, including the interchanging ones, is shared by all the set of health centres of the whole environment under study. In the case that not all the information of the subpopulations distributions at other patches are known by the health centre of each particular patch, the feedback vaccination rule would have a decentralized nature. The paper investigates the existence, allocation (depending on the vaccination control gains) and uniqueness of the disease-free equilibrium point as well as the existence of at least a stable endemic equilibrium point. Such a point coincides with the disease-free equilibrium point if the reproduction number is unity. The stability and instability of the disease-free equilibrium point are ensured under the values of the disease reproduction number guaranteeing, respectively, the un-attainability (the reproduction number being less than unity) and stability (the reproduction number being more than unity) of the endemic equilibrium point. The whole set of the potential endemic equilibrium points is characterized and a particular case is also described related to its uniqueness in the case when the patchy model reduces to a unique patch. Vaccination control laws including feedback are proposed which can take into account shared information between the various patches. It is not assumed that there are in the most general case, symmetry-type constrains on the population fluxes between the various patches or in the associated control gains parameterizations.
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