An epidemic disease caused by a new coronavirus has spread in Northern Italy with a strong contagion rate. We implement an SEIR model to compute the infected population and number of casualties of this epidemic. The example may ideally regard the situation in the Italian Region of Lombardy, where the epidemic started on February 25. We calibrate the model with the number of dead individuals to date (May 5, 2020) and constraint the parameters on the basis of values reported in the literature. The peak occurs at day 37 (March 31) approximately, when there is a rapid decrease, with a reproduction ratio R0 = 3 initially, 1.36 at day 22 and 0.8 after day 35, indicating different degrees of lockdown. The predicted death toll is approximately 15600 casualties, with 2.7 million infected individuals at the end of the epidemic. The incubation period providing a better fit of the dead individuals is 4.25 days and the infection period is 4 days, with a fatality rate of 0.00144/day [values based on the reported (official) number of casualties]. The infection fatality rate (IFR) is 0.57 %, and 2.36 % if twice the reported number of casualties is assumed. However, these rates depend on the initially exposed individuals. If approximately nine times more individuals are exposed, there are three times more infected people at the end of the epidemic and IFR = 0.47 %. If we relax these constraints and use a wider range of lower and upper bounds for the incubation and infection periods, we observe that a higher incubation period (13 versus 4.25 days) gives the same IFR (0.6 versus 0.57 %), but nine times more exposed individuals in the first case. Therefore, a precise determination of the fatality rate is subject to the knowledge of the characteristics of the epidemic.
We analytically study the SEIR (Susceptible Exposed Infectious Removed) epidemic model. The aim is to provide simple analytical expressions for the peak and asymptotic values and their characteristic times of the populations affected by the COVID-19 pandemic.
An analytic solution is obtained to the SEIR Epidemic Model. The solution is created by constructing a single second-order nonlinear differential equation in $\ln S$ and analytically continuing its divergent power series solution such that it matches the correct long-time exponential damping of the epidemic model. This is achieved through an asymptotic approximant (Barlow et. al, 2017, Q. Jl Mech. Appl. Math, 70 (1), 21-48) in the form of a modified symmetric Padé approximant that incorporates this damping. The utility of the analytical form is demonstrated through its application to the COVID…
The primary goal of this research is to investigate the impact of delay on the dynamics of the Susceptible-Exposed-Infected-Recovered-Death and Susceptible (SEIRDS) model, to which we add a stochastic term to account for uncertainty in COVID-19 parameter estimations. We run two models, one deterministic and one stochastic, and show that their solutions exist and are unique. We also numerically investigate the impact of immunity loss on the emerging time of a new wave, as well as the necessary condition for the extinction and persistence of the disease.
In this work, we have developed a Coxian distributed SEIR model in incorporating an empirical incubation period. We show that the global dynamics are completely determined by a basic reproduction number. An application of the Coxian distributed SEIR model using data of an empirical incubation period is explored. The model may be useful for resolving causing the realistic intrinsic parts in classical epidemic models since Coxian distribution approximately converges to any distribution.
A pandemic caused by a new coronavirus has spread worldwide, affecting Argentina. We implement an SEIR model to analyze the disease evolution in Buenos Aires and neighbouring cities. The model parameters are calibrated using the number of casualties officially reported. Since infinite solutions honour the data, we show different cases. In all of them the reproduction ratio $R_0$ decreases after early lockdown, but then raises, probably due to an increase in contagion in highly populated slums. Therefore it is mandatory to reverse this growing trend in $R_0$ by applying control strategies to av…
The infections and fatalities due to SARS-CoV-2 virus for cases specific to India have been studied using a deterministic susceptible-exposed-infected-recovered-dead (SEIRD) compartmental model. One of the most significant epidemiological parameter, namely the effective reproduction number of the infection is extracted from the daily growth rate data of reported infections and it is included in the model with a time variation. We evaluate the effect of control interventions implemented till now and estimate the case numbers for infections and deaths averted by these restrictive measures. We fu…
We propose an SEIR model for the populations and an SEI model for the vector to describe the transmission dynamics of a four-strain model with both primary and secondary dengue infections. In order to accomplish this, we propose and obtain an analytic solution of a system of 47 coupled differential equations. This would be the most complete epidemic model proposed to describe the dengue epidemic.
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…
COVID-19 has forced quarantine measures in several countries across the world. These measures have proven to be effective in significantly reducing the prevalence of the virus. To date, no effective treatment or vaccine is available. In the effort of preserving both public health as well as the economical and social textures, France and Italy governments have partially released lockdown measures. Here we extrapolate the long-term behavior of the epidemics in both countries using a Susceptible-Exposed-Infected-Recovered (SEIR) model where parameters are stochastically perturbed to handle the un…
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 …
Exact solutions of the SEIR epidemic model are derived, and various properties of solutions are obtained directly from the exact solution. In this paper Abel differential equations play an important role in establishing the exact solution of SEIR differential system, in particular the number of infected individuals can be represented in a simple form by using a positive solution of an Abel differential equation. It is shown that the parametric form of the exact solution satisfies some linear differential system including a positive solution of an Abel differential equation.
In this paper we analyze the effects of commuting and social inequalities for the epidemic development of the novel coronavirus (COVID-19). With this aim we consider a SEIRD (susceptible, exposed, infected, recovered and dead by disease) model without vital dynamics in a population divided into patches that have different economic resources and in which the individuals can commute from one patch to another (bilaterally). In the modeling we choose the social and commuting parameters arbitrarily. We calculate the basic reproductive number $R_0$ with the next generation approach and analyze the s…
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, …
Contact tracing and quarantine are well established non-pharmaceutical epidemic control tools. The paper aims to clarify the impact of these measures in COVID-19 epidemic. A new deterministic model is introduced (SEIRQ: susceptible, exposed, infectious, removed, quarantined) with Q compartment capturing individuals and releasing them with delay. We obtain a simple rule defining the reproduction number $\mathcal{R}$ in terms of quarantine parameters, ratio of diagnosed cases and transmission parameters. The model is applied to the epidemic in Poland in March - April 2020, when social distancing…
In this paper, I study epidemic diffusion in a generalized spatial SEIRD model, where individuals are initially connected in a social or geographical network. As the virus spreads in the network, the structure of interactions between people may endogenously change over time, due to quarantining measures and/or spatial-distancing policies. I explore via simulations the dynamic properties of the co-evolutionary process dynamically linking disease diffusion and network properties. Results suggest that, in order to predict how epidemic phenomena evolve in networked populations, it is not enough to…
A pandemic caused by a new coronavirus (COVID-19) has spread worldwide, inducing an epidemic still active in Argentina. In this chapter, we present a case study using an SEIR (Susceptible-Exposed-Infected-Recovered) diffusion model of fractional order in time to analyze the evolution of the epidemic in Buenos Aires and neighboring areas (Región Metropolitana de Buenos Aires, (RMBA)) comprising about 15 million inhabitants. In the SEIR model, individuals are divided into four classes, namely, susceptible (S), exposed (E), infected (I) and recovered (R). The SEIR model of fractional order allows…
We consider the $SEIRS$ epidemiology model with such features of the COVID-19 outbreak as: abundance of unidentified infected individuals, limited time of immunity and a possibility of vaccination. Within a compartmental realization of this model, we found the disease-free and the endemic stationary states. They exist in their respective restricted regions of the le via linear stability analysis. The expression for the basic reproductive number is obtained as well. The positions and heights of a first peak for the fractions of infected individuals are obtained numerically and are fitted to sim…
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 …
We study the dynamic evolution of COVID-19 cased by the Omicron variant via a fractional susceptible-exposedinfected-removed (SEIR) model. Preliminary data suggest that the symptoms of Omicron infection are not prominent and the transmission is therefore more concealed, which causes a relatively slow increase in the detected cases of the new infected at the beginning of the pandemic. To characterize the specific dynamics, the Caputo-Hadamard fractional derivative is adopted to refined the classical SEIR model. Based on the reported data, we infer the fractional order, timedependent parameters,…
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…
We measure the effect of different public health regulations to the spread of COVID-19, based on a SEIRA model -- a SEIR model including asymptomatic transmissions. The cumulative confirmed cases and death show nonlinear positive relationship with the value of asymptomatic rate. Based on this model, we analyze the inhibit effects to COVID-19 of three types of public health policies, i.e. isolation of laboratory confirmed cases, general personal protection and quarantine (lock-down). The simulations conclude that the isolation display limited effects to the asymptomatic viral carriers. The gene…
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 …
Recent evidences show that individuals who recovered from COVID-19 can be reinfected. However, this phenomenon has rarely been studied using mathematical models. In this paper, we propose a SEIRE epidemic model to describe the spread of the epidemic with reinfection. We obtain the important thresholds $R_0$ (the basic reproduction number) and Rc (a threshold less than one). Our investigations show that when $R_0 > 1$, the system has an endemic equilibrium, which is globally asymptotically stable. When $R_c < R_0 < 1$, the epidemic system exhibits bistable dynamics. That is, the system has back…
We present a class of SEIR Markov chain models for infectious diseases observed over discrete time in a random human population living in a closed environment. The population changes over time through random births, deaths, and transitions between states of the population. The SEIR models consist of random dynamical equations for each state (S, E, I and R) involving driving events for the process. We characterize some special types of SEIR Markov chain models in the class including: (1) when birth and death are zero or non-zero, and (2) when the incubation and infectious periods are constant o…
This paper is an exploratory study of two epidemiological questions on a worldwide basis. How fast is the disease spreading? Are the restrictions (especially mobility restrictions) for people bring the expected effect? To answer the first question, we propose a tool for estimating the reproduction number of epidemic (the number of secondary infections $R_t$) based on the SEIR model and compare it with an non-model $R_t$ estimation. To measure the $R_t$ of COVID-19 for different countries, real-time data on coronavirus daily cases of infections, recoveries, deaths are retrieved from the Center …
The Susceptible-Exposed-Infected-Recovered (SEIR) epidemiological model is one of the standard models of disease spreading. Here we analyse an extended SEIR model that accounts for asymptomatic carriers, believed to play an important role in COVID-19 transmission. For this model we derive a number of analytic results for important quantities such as the peak number of infections, the time taken to reach the peak and the size of the final affected population. We also propose an accurate way of specifying initial conditions for the numerics (from insufficient data) using the fact that the early …
We study the global dynamics of a reaction-diffusion SEIR infection model with distributed delay and nonlinear incidence rate. The well-posedness of the proposed model is proved. By means of Lyapunov functionals, we show that the disease free equilibrium state is globally asymptotically stable when the basic reproduction number is less or equal than one, and that the disease endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than one. Numerical simulations are provided to illustrate the obtained theoretical results.
The COVID-19 pandemic has influenced the lives of people globally. In the past year many researchers have proposed different models and approaches to explore in what ways the spread of the disease could be mitigated. One of the models that have been used a great deal is the Susceptible-Exposed-Infectious-Recovered (SEIR) model. Some researchers have modified the traditional SEIR model, and proposed new versions of it. However, to the best of our knowledge, the state-of-the-art papers have not considered the effect of different vaccine types, meaning single shot and double shot vaccines, in the…
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 …
The Susceptible-Exposed-Infectious-Recovered (SEIR) model is applied in several countries to ascertain the spread of the coronavirus disease 2019 (COVID-19). We consider discrete-time SEIR epidemic model in a closed system which does not account for births or deaths, total population size under consideration is constant. This dynamical system generated by a non-linear evolution operator depending on four parameters. Under some conditions on parameters we reduce the evolution operator to a quadratic stochastic operator (QSO) which maps 3-dimensional simplex to itself. We show that the QSO has u…
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…
In this paper we explore a time-depended SEIR model, in which the dynamics of the infection in four groups from a selected target group (population), divided according to the infection, are modeled by a system of nonlinear ordinary differential equations. Several basic parameters are involved in the model: coefficients of infection rate, incubation rate, recovery rate. The coefficients are adaptable to each specific infection, for each individual country, and depend on the measures to limit the spread of the infection and the effectiveness of the methods of treatment of the infected people in …
Compartmental transmission models have become an invaluable tool to study the dynamics of infectious diseases. The Susceptible-Infectious-Recovered (SIR) model is known to have an exact semi-analytical solution. In the current study, the approach of Harko et al. (2014) is generalised to obtain an approximate semi-analytical solution of the Susceptible-Exposed-Infectious-Recovered (SEIR) model. The SEIR model curves have nearly the same shapes as the SIR ones, but with a stretch factor applied to them across time that is related to the ratio of the incubation to infectious periods. This finding…
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 …
A reaction-diffusion model was developed describing the spread of the COVID-19 virus considering the mean daily movement of susceptible, exposed and asymptomatic individuals. The model was calibrated using data on the confirmed infection and death from France as well as their initial spatial distribution. First, the system of partial differential equations is studied, then the basic reproduction number, R0 is derived. Second, numerical simulations, based on a combination of level-set and finite differences, shown the spatial spread of COVID-19 from March 16 to June 16. Finally, scenarios of un…
We consider the SEIRS compartment epidemiology model suitable for predicting the evolution of the COVID-19 pandemy in the extreme limiting case of no acquired immunity. The disease-free and endemic fixed points are found and their stability is analysed. The expression for the basic reproduction ratio is obtained and discussed, emphasizing on its dependence on the model parameters. The threshold contact ratio is found which determines the possibility for a stable disease-free fixed point existence. Numeric solution for the pandemy evolution is also undertaken together with the approximate analy…
COVID-19 epidemics in Japan and Tokyo were analyzed by a fundamental equation of the dynamic phase transition. As a result, the epidemic was found to be in good agreement with the random nucleation and linear growth model suggesting that the epidemic between March 13, 2020 and May 22, 2020 was simply rate-limited by the three constant-parameters: the initial susceptible, domain growth rate, and nucleation decay constant. This model provides a good predictor of the epidemic because it consists of one equation and the initial specific plot is linear.
We present an early version of a Susceptible-Exposed-Infected-Recovered-Deceased (SEIRD) mathematical model based on partial differential equations coupled with a heterogeneous diffusion model. The model describes the spatio-temporal spread of the COVID-19 pandemic, and aims to capture dynamics also based on human habits and geographical features. To test the model, we compare the outputs generated by a finite-element solver with measured data over the Italian region of Lombardy, which has been heavily impacted by this crisis between February and April 2020. Our results show a strong qualitati…
We study Susceptible-Exposed-Asymptomatic-Infectious-Recovered (SEAIR) epidemic spreading model of COVID-19. It captures two important characteristics of the infectiousness of COVID-19: delayed start and its appearance before onset of symptoms, or even with total absence of them. The model is theoretically analyzed in continuous-time compartmental version and discrete-time version on random regular graphs and complex networks. We show analytically that there are relationships between the epidemic thresholds and the equations for the susceptible populations at the endemic equilibrium in all thr…
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…
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, 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 …
Following the spread of the COVID-19 pandemic and pending the establishment of vaccination campaigns, several non pharmaceutical interventions such as partial and full lockdown, quarantine and measures of physical distancing have been imposed in order to reduce the spread of the disease and to lift the pressure on healthcare system. Mathematical models are important tools for estimating the impact of these interventions, for monitoring the current evolution of the epidemic at a national level and for estimating the potential long-term consequences of relaxation of measures. In this paper, we m…
We introduce an extended SEIR infectious disease model with data assimilation for the study of the spread of COVID-19. In this framework, undetected asymptomatic and pre-symptomatic cases are taken into account, and the impact of their uncertain proportion is fully investigated. The standard SEIR model does not consider these populations, while their role in the propagation of the disease is acknowledged. An ensemble Kalman filter is implemented to assimilate reliable observations of three compartments in the model. The system tracks the evolution of the effective reproduction number and estim…
The present article studies the extension of two deterministic models for describing the novel coronavirus pandemic crisis, the SIR model and the SEIR model. The models were studied and compared to real data in order to support the validity of each description and extract important information regarding the pandemic, such as the basic reproductive number R0, which might provide useful information concerning the rate of increase of the pandemic predicted by each model. We next proceed to making predictions and comparing more complex models derived from the SEIR model with the SIRD model, in ord…
Mexico, like the rest of the world, is currently facing the The COVID-19 pandemic. Given the size of its territory, the efforts to contain the disease have involved both national and regional measures. For this work, the curves of daily new cases of each municipality reported by the federal government were compared. We found that 114 municipalities form a large network of statistically dependent epidemic phenomena. Based on the network's modular structure, these 114 municipalities can be split into four distinct communities of coordinated epidemic phenomena. These clusters are not limited by g…
While COVID-19 is rapidly propagating around the globe, the need for providing real-time forecasts of the epidemics pushes fits of dynamical and statistical models to available data beyond their capabilities. Here we focus on statistical predictions of COVID-19 infections performed by fitting asymptotic distributions to actual data. By taking as a case-study the epidemic evolution of total COVID-19 infections in Chinese provinces and Italian regions, we find that predictions are characterized by large uncertainties at the early stages of the epidemic growth. Those uncertainties significantly r…
The aim of this paper consists in the application of a recent epidemiological model, namely SEIR with Social Distancing (SEIR--SD), extended here through the definition of a social distancing function varying over time, to assess the situation related to the spreading of the coronavirus Covid--19 in Italy and in two of its most important regions, i.e., Lombardy and Campania. To profitably use this model, the most suitable values of its parameters must be found. The estimation of the SEIR--SD model parameters takes place here through the use of Differential Evolution, a heuristic optimization t…
In this study, we analyze the effectiveness of measures aimed at finding and isolating infected individuals to contain epidemics like COVID-19, as the suppression induced over the effective reproduction number. We develop a mathematical model to compute the relative suppression of the effective reproduction number of an epidemic that such measures produce. This outcome is expressed as a function of a small set of parameters that describe the main features of the epidemic and summarize the effectiveness of the isolation measures. In particular, we focus on the impact when a fraction of the popu…
In this paper, we propose a new real-time differential virus transmission model, which can give more accurate and robust short-term predictions of COVID-19 transmitted infectious disease with benefits of near-term trend projection. Different from the existing Susceptible-Exposed-Infected-Removed (SEIR) based virus transmission models, which fits well for pandemic modeling with sufficient historical data, the new model, which is also SEIR based, uses short history data to find the trend of the changing disease dynamics for the infected, the dead and the recovered so that it can naturally accomm…
A mathematical model was developed describing the dynamic of the COVID-19 virus over a population considering that the infected can either be symptomatic or not. The model was calibrated using data on the confirmed cases and death from several countries like France, Philippines, Italy, Spain, United Kingdom, China, and the USA. First, we derived the basic reproduction number, $\mathcal{R}_0$, and estimated the effective reproduction $\mathcal{R}_{\rm eff}$ for each country. Second, we were interested in the merits of interventions, either by distancing or by treatment. Results revealed that to…
