Why can we observe a plateau even in an out of control epidemic outbreak? A SEIR model with the interaction of $n$ distinct populations for Covid-19 in Brazil
Matheus Jatkoske Lazo, Adriano De Cezaro
备注：
submitted to Trends in Applied and Computational Mathematics
This manuscript proposes a model of $n$ distinct populations interaction structured SEIR to describe the spread of COVID-19 pandemic diseases. The proposed model has the flexibility to include geographically separated communities as well as taking into account aging population groups and their interactions. We show that distinct assumptions on the dynamics of the proposed model lead to a plateau-like curve of the infected population, reflecting collected data from large countries like Brazil. Such observations aim to the following conjecture: " Covid-19 diseased diffusion from the capitals to the Brazil interior, as reflected by the proposed model, is responsible for plateau-like reported cases in the country". We present numerical simulations of some scenarios and comparison with reported data from Brazil that corroborate with the aforementioned conclusions.
We shall apply a generalized SEIR model to study the outbreak of COVID-19 in Brazil. In particular, we would like to explain the projections of the increase in the level of infection over a long period of time, overlapping large local outbreaks in the most populous states in the country. A time-dependent dynamic SEIR model inspired in a model previously used during the outbreak in China was used to analyses the time trajectories of infected, recovered, and deaths. The model has parameters that vary with time and are fitted considering a nonlinear least-squares method. The simulations starting …
The Covid-19 outbreak of 2020 has required many governments to develop mathematical-statistical models of the outbreak for policy and planning purposes. This work provides a tutorial on building a compartmental model using Susceptibles, Exposed, Infected, Recovered and Deaths status through time. A Bayesian Framework is utilized to perform both parameter estimation and predictions. This model uses interventions to quantify the impact of various government attempts to slow the spread of the virus. Predictions are also made to determine when the peak Active Infections will occur.
This article contains a series of analyses done for the SARS-CoV-2 outbreak in Rio Grande do Sul (RS) in the south of Brazil. These analyses are focused on the high-incidence cities such as the state capital Porto Alegre and at the state level. We provide methodological details and estimates for the effective reproduction number $R_t$, a joint analysis of the mobility data together with the estimated $R_t$ as well as ICU simulations and ICU LoS (length of stay) estimation for hospitalizations in Porto Alegre/RS.
COVID-19 pandemic has become a major threat to the country. Till date, well tested medication or antidote is not available to cure this disease. According to WHO reports, COVID-19 is a severe acute respiratory syndrome which is transmitted through respiratory droplets and contact routes. Analysis of this disease requires major attention by the Government to take necessary steps in reducing the effect of this global pandemic. In this study, outbreak of this disease has been analysed for India till 30th March 2020 and predictions have been made for the number of cases for the next 2 weeks. SEIR …
Background: The analysis of the Sars-CoV-2 epidemic is of paramount importance to understand the dynamics of the coronavirus spread. This can help health and government authorities take the appropriate measures and implement suitable politics aimed at fighting and preventing it. Methods: A time-dependent dynamic SEIR model inspired in a model previously used during the MERS outbreak in South Korea was used to analyse the time trajectories of active and hospitalized cases in Portugal. Results: The time evolution of the virus spread in the country was adequately modelled. The model has changeabl…
Multiple instances of Zika virus epidemic have been reported around the world in the last two decades, turning the related illness into an international concern. In this context the use of mathematical models for epidemics is of great importance, since they are useful tools to study the underlying outbreak numbers and allow one to test the effectiveness of different strategies used to combat the associated diseases. This work deals with the development and calibration of an epidemic model to describe the 2016 outbreak of Zika virus in Brazil. A system of 8 differential equations with 8 paramet…
We study variants of the SEIR model for interpreting some qualitative features of the statistics of the Covid-19 epidemic in France. Standard SEIR models distinguish essentially two regimes: either the disease is controlled and the number of infected people rapidly decreases, or the disease spreads and contaminates a significant fraction of the population until herd immunity is achieved. After lockdown, at first sight it seems that social distancing is not enough to control the outbreak. We discuss here a possible explanation, namely that the lockdown is creating social heterogeneity: even if …
In this paper we consider a SEIRD epidemic model for a population composed by two groups with asymmetric interaction. Given two sets of parameters for the model, we estimate the distance of the solutions for the second group based on the distance of solutions for the first one. We also study the final size of the epidemic for each group. We illustrate our results with the spread of coronavirus disease 2019 (COVID-19) pandemic in the New York County (USA) for the initial stage of the contamination, and in the the cities of Petrolina and Juazeiro (Brazil).
An SEIRS epidemic with disease fatalities is introduced in a growing population (modelled as a super-critical linear birth and death process). The study of the initial phase of the epidemic is stochastic, while the analysis of the major outbreaks is deterministic. Depending on the values of the parameters, the following scenarios are possible. i) The disease dies out quickly, only infecting few; ii) the epidemic takes off, the \textit{number} of infected individuals grows exponentially, but the \textit{fraction} of infected individuals remains negligible; iii) the epidemic takes off, the \text…
We consider the SIR epidemiological model applied to the evolution of COVID-19 with two approaches. In the first place we fit a global SIR model, with time delay, and constant parameters throughout the outbreak, including the contagion rate. The contention measures are reflected on an effective reduced susceptible population $N_{eff}$. In the second approach we consider a time-dependent contagion rate that reflects the contention measures either through a step by step fitting process or by following an exponential decay. In this last model the population is considered the one of the country $N…
The coronavirus disease 2019 (COVID-19) had caused more that 8 million infections as of middle June 2020. Recently, Brazil has become a new epicentre of COVID-19, while India and African region are potential epicentres. This study aims to predict the inflection point and outbreak size of these new/potential epicentres at the early phase of the epidemics by borrowing information from more `mature' curves from other countries. We modeled the cumulative cases to the well-known sigmoid growth curves to describe the epidemic trends under the mixed-effect models and using the four-parameter logistic…
In this paper, a generalized fractional-order SEIR model is proposed, denoted by SEIQRP model, which has a basic guiding significance for the prediction of the possible outbreak of infectious diseases like COVID-19 and other insect diseases in the future. Firstly, some qualitative properties of the model are analyzed. The basic reproduction number $R_{0}$ is derived. When $R_{0}<1$, the disease-free equilibrium point is unique and locally asymptotically stable. When $R_{0}>1$, the endemic equilibrium point is also unique. Furthermore, some conditions are established to ensure the local asympto…
In this paper, we introduce a novel modeling framework for incorporating fear of infection and frustration with social distancing into disease dynamics. We show that the resulting SEIR behavior-perception model has three principal modes of qualitative behavior---no outbreak, controlled outbreak, and uncontrolled outbreak. We also demonstrate that the model can produce transient and sustained waves of infection consistent with secondary outbreaks. We fit the model to cumulative COVID-19 case and mortality data from several regions. Our analysis suggests that regions which experience a significa…
First highly pathogenic avian influenza outbreak in a commercial farm in Brazil: outbreak timeline, control actions, risk analysis, and transmission modeling
North Carolina State University（北卡罗来纳州立大学）；Departamento de Defesa Agropecuária, Secretaria da Agricultura, Pecuária e Desenvolvimento Rural（农业、畜牧业和农村发展局农业防御部）；Polytechnic College of Federal University of Santa Maria（圣玛丽亚联邦大学理工学院）
The paper deals with a modeling of the ongoing epidemic caused by Coronavirus disease 2019 (COVID-19) on the closed territory of the Republic of Croatia. Using the official public information on the number of confirmed infected, recovered and deceased individuals, the modified SEIR compartmental model is developed to describe the underlying dynamics of the epidemic. Fitted modified SEIR model provides the prediction of the disease progression in the near future, considering strict control interventions by means of social distancing and quarantine for infected and at-risk individuals introduced…
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In this paper we propose a Susceptible-Infected-Exposed-Recovered-Dead (SEIRD) differential model for the analysis and forecast of the COVID-19 spread in some regions of Italy, using the data from the Italian Protezione Civile from February 24th 2020. In this study investigate an adaptation of the model. Since several restricting measures have been imposed by the Italian government at different times, starting from March 8th 2020, we propose a modification of SEIRD by introducing a time dependent transmitting rate. In the numerical results we report the maximum infection spread for the three I…
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The COVID-19 pandemic poses challenges for continuing economic activity while reducing health risks. While these challenges can be mitigated through testing, testing budget is often limited. Here we study how institutions, such as nursing homes, should utilize a fixed test budget for early detection of an outbreak. Using an extended network-SEIR model, we show that given a certain budget of tests, it is generally better to test smaller subgroups of the population frequently than to test larger groups but less frequently. The numerical results are consistent with an analytical expression we der…
The 2019 Novel Corona virus infection (COVID 19) is an ongoing public health emergency of international focus. Significant gaps persist in our knowledge of COVID 19 epidemiology, transmission dynamics, investigation tools and management, despite (or possibly because of) the fact that the outbreak is an unprecedented global threat. On the positive side, enough is currently known about the epidemic process to permit the construction of mathematical predictive models. In our work, we adapt a traditional SEIR epidemic model to the specific dynamic compartments and epidemic parameters of COVID 19, …
In this study, we propose a time-dependent Susceptible-Exposed-Infected-Recovered (SEIR) model for the analysis of the SARS-CoV-2 epidemic outbreak in three different countries, the United States of America, Italy and Iceland using public data inherent the numbers of the epidemic wave. Since several types and grades of actions were adopted by the governments, including travel restrictions, social distancing, or limitation of movement, we want to investigate how these measures can affect the epidemic curve of the infectious population. The parameters of interest for the SEIR model were estimate…
The outbreak and propagation of COVID-19 have posed a considerable challenge to modern society. In particular, the different restrictive actions taken by governments to prevent the spread of the virus have changed the way humans interact and conceive interaction. Due to geographical, behavioral, or economic factors, different sub-groups among a population are more (or less) likely to interact, and thus to spread/acquire the virus. In this work, we present a general multi-group SEIRA model for representing the spread of COVID-19 among a heterogeneous population and test it in a numerical case o…
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Ebola virus is one of the most virulent pathogens for humans. We present a mathematical description of different Susceptible-Exposed-Infectious-Recovered (SEIR) models. By using mathematical modeling and analysis, the latest major outbreak of Ebola virus in West Africa is described. Our aim is to study and discuss the properties of SEIR models with respect to Ebola virus, the information they provide, and when the models make sense. We added to the basic SEIR model demographic effects in order to analyze the equilibria with vital dynamics. Numerical simulations confirm the theoretical analysis…
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OBJECTIVES: to describe the first wave of the COVID-19 pandemic with a focus on undetected cases and to evaluate different post-lockdown scenarios. DESIGN: the study introduces a SEIR compartmental model, taking into account the region-specific fraction of undetected cases, the effects of mobility restrictions, and the personal protective measures adopted, such as wearing a mask and washing hands frequently. SETTING AND PARTICIPANTS: the model is experimentally validated with data of all the Italian regions, some European countries, and the US. MAIN OUTCOME MEASURES: the accuracy of the model …
In the absence of other tools, monitoring the effects of protective measures, including social distancing and forecasting the outcome of outbreaks is of immense interest. Real-time data is noisy and very often hampered by systematic errors in reporting. Detailed epidemic models may contain a large number of empirical parameters, which cannot be determined with sufficient accuracy. In this paper, we show that the cumulative number of deaths can be regarded as a master variable, and the parameters of the epidemic such as the basic reproduction number, the size of the susceptible population, and …
The outbreak of COVID-19 in 2020 has led to a surge in interest in the mathematical modeling of infectious diseases. Such models are usually defined as compartmental models, in which the population under study is divided into compartments based on qualitative characteristics, with different assumptions about the nature and rate of transfer across compartments. Though most commonly formulated as ordinary differential equation (ODE) models, in which the compartments depend only on time, recent works have also focused on partial differential equation (PDE) models, incorporating the variation of a…
This paper introduces a new multi-strain epidemic model with reinfection and cross-immunity to provide insights into the resurgence of the COVID-19 epidemic in an area with reportedly high seroprevalence due to a largely unmitigated outbreak: the state of Amazonas, Brazil. Although high seroprevalence could have been expected to trigger herd immunity and prevent further waves in the state, we have observed persistent levels of infection after the first wave and eventually the emergence of a second viral strain just before an augmented second wave. Our experiments suggest that the persistent le…
Without vaccines and treatments, societies must rely on non-pharmaceutical intervention strategies to control the spread of emerging diseases such as COVID-19. Though complete lockdown is epidemiologically effective, because it eliminates infectious contacts, it comes with significant costs. Several recent studies have suggested that a plausible compromise strategy for minimizing epidemic risk is periodic closure, in which populations oscillate between wide-spread social restrictions and relaxation. However, no underlying theory has been proposed to predict and explain optimal closure periods …
Mathematical models are traditionally used to analyze the long-term global evolution of epidemics, to determine the potential and severity of an outbreak, and to provide critical information for identifying the type of disease interventions and intensity. One of the widely used mathematical models of long-term spreading of epidemics are the so-called deterministic compartmental models (SIR/SEIR type models). One of the main purposes of applying such models is to assess how the expensive restriction measures imposed by the authorities (home and social isolation/quarantine, travel restrictions, …
In this paper, based on the Akaike information criterion, root mean square error and robustness coefficient, a rational evaluation of various epidemic models/methods, including seven empirical functions, four statistical inference methods and five dynamical models, on their forecasting abilities is carried out. With respect to the outbreak data of COVID-19 epidemics in China, we find that before the inflection point, all models fail to make a reliable prediction. The Logistic function consistently underestimates the final epidemic size, while the Gompertz's function makes an overestimation in …
Assessing the performance of compartmental and renewal models for learning $R_{t}$ using spatially heterogeneous epidemic simulations on real geographies
We introduce a dengue model (SEIR) where the human individuals are treated on an individual basis (IBM) while the mosquito population, produced by an independent model, is treated by compartments (SEI). We study the spread of epidemics by the sole action of the mosquito. Exponential, deterministic and experimental distributions for the (human) exposed period are considered in two weather scenarios, one corresponding to temperate climate and the other to tropical climate. Virus circulation, final epidemic size and duration of outbreaks are considered showing that the results present little sens…
After their re-emergence in the last decades, dengue fever and other vector-borne diseases are a potential threat to the lives of millions of people. Based on a data set of dengue cases in the Brazilian city of Fortaleza, collected from 2011 to 2016, we study the spatio-temporal characteristics of dengue outbreaks to characterize epidemic and non-epidemic years. First, we identify regions that show a high prevalence of dengue cases and mosquito larvae in different years and also analyze their corresponding correlations. Our results show that the characteristic correlation length of the epidemi…
The outbreak of Covid-19 led the world to an unprecedent health and economical crisis. In an attempt to responde to this emergency researchers worldwide are intensively studying the Covid-19 pandemic dynamics. In this work, a SIRSi compartmental model is proposed, which is a modification of the known classical SIR model. The proposed SIRSi model considers differences in the immunization within a population, and the possibility of unreported or asymptomatic cases. The model is adjusted to three major cities of São Paulo State, in Brazil, namely, São Paulo, Santos and Campinas, providing estimat…
The outbreak of COVID-19 in 2020 has led to a surge in the interest in the mathematical modeling of infectious diseases. Disease transmission may be modeled as compartmental models, in which the population under study is divided into compartments and has assumptions about the nature and time rate of transfer from one compartment to another. Usually, they are composed of a system of ordinary differential equations (ODE's) in time. A class of such models considers the Susceptible, Exposed, Infected, Recovered, and Deceased populations, the SEIRD model. However, these models do not always account…
The goal of this work is to study the optimal controls for the COVID-19 epidemic in Brazil. We consider an age-structured SEIRQ model with quarantine compartment, where the controls are the quarantine entrance parameters. We then compare the optimal controls for different quarantine lengths and distribution of the total control cost by assessing their respective reductions in deaths in comparison to the same period without quarantine. The best strategy provides a calendar of when to relax the isolation measures for each age group. Finally, we analyse how a delay in the beginning of the quarant…
We study implications of the dynamical and spatial contact structure between Brazilian escorts and sex-buyers for the spreading of sexually transmitted infections (STI). Despite a highly skewed degree distribution diseases spreading in this contact structure have rather well-defined epidemic thresholds. Temporal effects create a broad distribution of outbreak sizes even if the transmission probability is taken to the hypothetical value of 100%. Temporal correlations speed up outbreaks, especially in the early phase, compared to randomized contact structures. The time-ordering and the network t…
The goal of this work is to analyse the effects of control policies for the coronavirus (COVID-19) epidemic in Brazil. This is done by considering an age-structured SEIR model with a quarantine class and two types of controls. The first one studies the sensitivity with regard to the parameters of the basic reproductive number R0 which is calculated by the next generation method. The second one evaluates different quarantine strategies by comparing their relative total number of deaths.
Mathematical models are widely recognized as an important tool for analyzing and understanding the dynamics of infectious disease outbreaks, predict their future trends, and evaluate public health intervention measures for disease control and elimination. We propose a novel stochastic metapopulation state-space model for COVID-19 transmission, based on a discrete-time spatio-temporal susceptible/exposed/infected/recovered/deceased (SEIRD) model. The proposed framework allows the hidden SEIRD states and unknown transmission parameters to be estimated from noisy, incomplete time series of report…
With the unfolding of the COVID-19 pandemic, mathematical modeling of epidemics has been perceived and used as a central element in understanding, predicting, and governing the pandemic event. However, soon it became clear that long term predictions were extremely challenging to address. Moreover, it is still unclear which metric shall be used for a global description of the evolution of the outbreaks. Yet a robust modeling of pandemic dynamics and a consistent choice of the transmission metric is crucial for an in-depth understanding of the macroscopic phenomenology and better-informed mitiga…
The current outbreak of the coronavirus disease 2019 (COVID-19) is an unprecedented example of how fast an infectious disease can spread around the globe (especially in urban areas) and the enormous impact it causes on public health and socio-economic activities. Despite the recent surge of investigations about different aspects of the COVID-19 pandemic, we still know little about the effects of city size on the propagation of this disease in urban areas. Here we investigate how the number of cases and deaths by COVID-19 scale with the population of Brazilian cities. Our results indicate small…
The goal of this study is to establish a general framework for predicting the so-called critical Turning Period in an infectious disease epidemic such as the COVID-19 outbreak in China early this year. This framework enabled a timely prediction of the turning period when applied to Wuhan COVID-19 epidemic and informed the relevant authority for taking appropriate and timely actions to control the epidemic. It is expected to provide insightful information on turning period for the world's current battle against the COVID-19 pandemic. The underlying mathematical model in our framework is the ind…
HKUST (University of Science & Technology)（香港科技大学）；International Computer Science Institute（国际计算机科学研究所）；Ecole de Guerre Economique（经济战争学院）；University of Cambridge（剑桥大学）；Manifold Research；University College London（伦敦大学学院）；ELU AI Ltd（ELU AI有限公司）；Royal Free Hospital（皇家自由医院）；Population Research Institute, The Family Federation of Finland（芬兰家庭联合会人口研究所）
We present two models for the COVID-19 pandemic predicting the impact of universal face mask wearing upon the spread of the SARS-CoV-2 virus--one employing a stochastic dynamic network based compartmental SEIR (susceptible-exposed-infectious-recovered) approach, and the other employing individual ABM (agent-based modelling) Monte Carlo simulation--indicating (1) significant impact under (near) universal masking when at least 80% of a population is wearing masks, versus minimal impact when only 50% or less of the population is wearing masks, and (2) significant impact when universal masking is …
The goal of this work is to consider widespread use of face masks as a non-pharmaceutical control strategy for the Covid-19 pandemic. A SEIR model that divides the population into individuals that wear masks and those that do not is considered. After calculating the basic reproductive number by a next generation approach, a criterion for determining when an epidemic can be prevented by the use of masks only and the critical percentage of mask users for disease prevention in the population are derived. The results are then applied to real world data from the United States, Brazil and Italy.
This paper is concerned with stochastic SIR and SEIR epidemic models on random networks in which individuals may rewire away from infected neighbors at some rate $ω$ (and reconnect to non-infectious individuals with probability $α$ or else simply drop the edge if $α=0$), so-called preventive rewiring. The models are denoted SIR-$ω$ and SEIR-$ω$, and we focus attention on the early stages of an outbreak, where we derive expression for the basic reproduction number $R_0$ and the expected degree of the infectious nodes $E(D_I)$ using two different approximation approaches. The first approach appr…
Since early 2020, the world has been dealing with a raging pandemic outbreak: COVID-19. A year later, vaccines have become accessible, but in limited quantities, so that governments needed to devise a strategy to decide which part of the population to prioritize when assigning the available doses, and how to manage the interval between doses for multi-dose vaccines. In this paper, we present an optimization framework to address the dynamic double-dose vaccine allocation problem whereby the available vaccine doses must be administered to different age-groups to minimize specific societal object…
