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Open accessJournal ArticleDOI: 10.3390/MATH9060636

An Extended SEIR Model with Vaccination for Forecasting the COVID-19 Pandemic in Saudi Arabia Using an Ensemble Kalman Filter

02 Mar 2021-Vol. 9, Iss: 6, pp 636
Abstract: In this paper, an extended SEIR model with a vaccination compartment is proposed to simulate the novel coronavirus disease (COVID-19) spread in Saudi Arabia. The model considers seven stages of infection: susceptible (S), exposed (E), infectious (I), quarantined (Q), recovered (R), deaths (D), and vaccinated (V). Initially, a mathematical analysis is carried out to illustrate the non-negativity, boundedness, epidemic equilibrium, existence, and uniqueness of the endemic equilibrium, and the basic reproduction number of the proposed model. Such numerical models can be, however, subject to various sources of uncertainties, due to an imperfect description of the biological processes governing the disease spread, which may strongly limit their forecasting skills. A data assimilation method, mainly, the ensemble Kalman filter (EnKF), is then used to constrain the model outputs and its parameters with available data. We conduct joint state-parameters estimation experiments assimilating daily data into the proposed model using the EnKF in order to enhance the model’s forecasting skills. Starting from the estimated set of model parameters, we then conduct short-term predictions in order to assess the predicability range of the model. We apply the proposed assimilation system on real data sets from Saudi Arabia. The numerical results demonstrate the capability of the proposed model in achieving accurate prediction of the epidemic development up to two-week time scales. Finally, we investigate the effect of vaccination on the spread of the pandemic.

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24 results found


Open access
Geir Evensen1Institutions (1)
01 Apr 2003-
Abstract: The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

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Topics: Ensemble Kalman filter (67%)

2,975 Citations


Open accessPosted Content
Abstract: Infectious diseases are a threat for human health with tremendous impact on our society at large. The recent COVID-19 pandemic, caused by the SARS-CoV-2, is the latest example of a highly infectious disease ravaging the world, since late 2019. It is therefore imperative to develop efficient mathematical models, able to substantially curb the damages of a pandemic by unveiling disease spreading dynamics and symmetries. This will help inform (non)-pharmaceutical prevention strategies. For the reasons above we wrote this report that goes at the heart of mathematical modelling of infectious disease diffusion by simultaneously investigating the underlying microscopic dynamics in terms of percolation models, effective description via compartmental models and the employment of temporal symmetries naturally encoded in the mathematical language of critical phenomena. Our report reviews these approaches and determines their common denominators, relevant for theoretical epidemiology and its link to important concepts in theoretical physics. We show that the different frameworks exhibit common features such as criticality and self-similarity under time rescaling. These features are naturally encoded within the unifying field theoretical approach. The latter leads to an efficient description of the time evolution of the disease via a framework in which (near) time-dilation invariance is explicitly realised. As important test of the relevance of symmetries we show how to mathematically account for observed phenomena such as multi-wave dynamics. The models presented here are of immediate relevance for different realms of scientific enquiry from medical applications to the understanding of human behaviour. Our review offers novel perspectives on how to model, capture, organise and understand epidemiological data and disease dynamics for modelling real-world phenomena.

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6 Citations


Journal ArticleDOI: 10.1016/J.CHAOS.2021.111129
Abstract: In this work, we formulate a mathematical model with a non-integer order derivative to investigate typhoid fever transmission dynamics. To combat the spread of this disease in the human community, control measures like vaccination are included in the proposed model. We calculate the epidemiological threshold called the control reproduction number, R c , and perform the asymptotic stability of the typhoid-free equilibrium point. We prove that the typhoid-free equilibrium for both integer and non-integer models is locally and globally asymptotically stable whenever R c is less than one. We also prove that both models admit only one endemic equilibrium point which is globally asymptotically stable whenever R c > 1 and no endemic equilibrium point otherwise. This means that the backward bifurcation phenomenon does not occur. In absence of vaccination, R c is equal to the basic reproduction number R 0 . We found out that R c R 0 which means that vaccination can permit to decrease of the spread of typhoid fever in the human community. Using fixed point theory, we perform existence and uniqueness analysis of solutions of the fractional model. We use the Adams-Bashforth-Moulton method to construct a numerical scheme of the fractional model. We prove the stability of the proposed numerical scheme. To calibrate our model, we estimate model parameters on clinical data of Mbandjock District Hospital in the Centre Region of Cameroon, using the Non-linear Least-Square method. This permits us to find R c = 1.3722 , which means that we are in an endemic state (since R c > 1 ), and then to predict new cases of typhoid fever per month at Mbandjock in the next new year. To determine model parameters that are responsible for disease spread in the human community, we perform sensitivity analysis (SA). This analysis shows that the vaccination rate, the human-bacteria contact rate, as well as the recovery rate, are the most important parameters in the disease spread. To validate our analytical results, and to see the impact of some control measures in the spread of typhoid fever in the human community, as well as the impact of the fractional-order on typhoid transmission dynamics, we perform several numerical simulations.

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3 Citations


Open accessJournal ArticleDOI: 10.3390/SYM13060947
26 May 2021-Symmetry
Abstract: In many countries the COVID-19 pandemic seems to witness second and third waves with dire consequences on human lives and economies Given this situation the modeling of the transmission of the disease is still the subject of research with the ultimate goal of understanding the dynamics of the disease and assessing the efficacy of different mitigation strategies undertaken by the affected countries We propose a mathematical model for COVID-19 transmission The model is structured upon five classes: an individual can be susceptible, exposed, infectious, quarantined or removed The model is based on a nonlinear incidence rate, takes into account the influence of media on public behavior, and assumes the recovery rate to be dependent on the hospital-beds to population ratio A detailed analysis of the proposed model is carried out, including the existence and uniqueness of solutions, stability analysis of the disease-free equilibrium (symmetry) and sensitivity analysis We found that if the basic reproduction number is less than unity the system can exhibit Hopf and backward bifurcations for some range of parameters Numerical simulations using parameter values fitted to Saudi Arabia are carried out to support the theoretical proofs and to analyze the effects of hospital-beds to population ratio, quarantine, and media effects on the predicted nonlinear behavior

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Topics: Population (54%), Basic reproduction number (50%)

1 Citations


Open accessJournal ArticleDOI: 10.1016/J.RINP.2021.104629
02 Aug 2021-Results in physics
Abstract: In this paper, we investigate the stochastic nature of the COVID-19 temporal dynamics by generating a fractional-order dynamic model and a fractional-order-stochastic model. Initially, we considered the first and second vaccination doses as multiple vaccinations were initiated worldwide. The concerned models are then tested for the Saudi Arabia second virus wave, which is assumed to start on 1st March 2021. Four daily vaccination scenarios for the first and second dose are assumed for 100 days from the wave beginning. One of these scenarios is based on function optimization using the invasive weed optimization algorithm (IWO). After that, we numerically solve the established models using the fractional Euler method and the Euler-Murayama method. Finally, the obtained virus dynamics using the assumed scenarios and the real one started by the government are compared. The optimized scenario using the IWO effectively minimizes the predicted cumulative wave infections with a 4.4 % lower number of used vaccination doses.

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Topics: Stochastic modelling (51%)

References
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52 results found


Journal ArticleDOI: 10.1115/1.3662552
R. E. Kalman1Institutions (1)
Topics: Filtering problem (55%), Fast Kalman filter (54%), Recursive Bayesian estimation (51%) ... read more

22,129 Citations


Open accessJournal ArticleDOI: 10.1098/RSPA.1927.0118
Abstract: (1) One of the most striking features in the study of epidemics is the difficulty of finding a causal factor which appears to be adequate to account for the magnitude of the frequent epidemics of disease which visit almost every population. It was with a view to obtaining more insight regarding the effects of the various factors which govern the spread of contagious epidemics that the present investigation was undertaken. Reference may here be made to the work of Ross and Hudson (1915-17) in which the same problem is attacked. The problem is here carried to a further stage, and it is considered from a point of view which is in one sense more general. The problem may be summarised as follows: One (or more) infected person is introduced into a community of individuals, more or less susceptible to the disease in question. The disease spreads from the affected to the unaffected by contact infection. Each infected person runs through the course of his sickness, and finally is removed from the number of those who are sick, by recovery or by death. The chances of recovery or death vary from day to day during the course of his illness. The chances that the affected may convey infection to the unaffected are likewise dependent upon the stage of the sickness. As the epidemic spreads, the number of unaffected members of the community becomes reduced. Since the course of an epidemic is short compared with the life of an individual, the population may be considered as remaining constant, except in as far as it is modified by deaths due to the epidemic disease itself. In the course of time the epidemic may come to an end. One of the most important probems in epidemiology is to ascertain whether this termination occurs only when no susceptible individuals are left, or whether the interplay of the various factors of infectivity, recovery and mortality, may result in termination, whilst many susceptible individuals are still present in the unaffected population. It is difficult to treat this problem in its most general aspect. In the present communication discussion will be limited to the case in which all members of the community are initially equally susceptible to the disease, and it will be further assumed that complete immunity is conferred by a single infection.

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Topics: Population (54%), Epidemic model (53%), Economic epidemiology (51%) ... read more

7,409 Citations


Open access
01 Jan 1927-
Abstract: (1) One of the most striking features in the study of epidemics is the difficulty of finding a causal factor which appears to be adequate to account for the magnitude of the frequent epidemics of disease which visit almost every population. It was with a view to obtaining more insight regarding the effects of the various factors which govern the spread of contagious epidemics that the present investigation was undertaken. Reference may here be made to the work of Ross and Hudson (1915-17) in which the same problem is attacked. The problem is here carried to a further stage, and it is considered from a point of view which is in one sense more general. The problem may be summarised as follows: One (or more) infected person is introduced into a community of individuals, more or less susceptible to the disease in question. The disease spreads from the affected to the unaffected by contact infection. Each infected person runs through the course of his sickness, and finally is removed from the number of those who are sick, by recovery or by death. The chances of recovery or death vary from day to day during the course of his illness. The chances that the affected may convey infection to the unaffected are likewise dependent upon the stage of the sickness. As the epidemic spreads, the number of unaffected members of the community becomes reduced. Since the course of an epidemic is short compared with the life of an individual, the population may be considered as remaining constant, except in as far as it is modified by deaths due to the epidemic disease itself. In the course of time the epidemic may come to an end. One of the most important probems in epidemiology is to ascertain whether this termination occurs only when no susceptible individuals are left, or whether the interplay of the various factors of infectivity, recovery and mortality, may result in termination, whilst many susceptible individuals are still present in the unaffected population. It is difficult to treat this problem in its most general aspect. In the present communication discussion will be limited to the case in which all members of the community are initially equally susceptible to the disease, and it will be further assumed that complete immunity is conferred by a single infection.

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Topics: Population (54%)

6,435 Citations


Journal ArticleDOI: 10.1016/S0025-5564(02)00108-6
P. van den Driessche1, James Watmough2Institutions (2)
Abstract: A precise definition of the basic reproduction number, Ro, is presented for a general compartmental disease transmission model based on a system of ordinary dierential equations. It is shown that, if Ro 1, then it is unstable. Thus,Ro is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for Ro near one. This criterion, together with the definition of Ro, is illustrated by treatment, multigroup, staged progression, multistrain and vectorhost models and can be applied to more complex models. The results are significant for disease control.

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5,559 Citations


Open accessJournal ArticleDOI: 10.1056/NEJMOA2034577
Abstract: Background Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection and the resulting coronavirus disease 2019 (Covid-19) have afflicted tens of millions of people in a world...

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4,222 Citations