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SEAIR Epidemic spreading model of COVID-19
Abstract
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 three versions, which hold when the epidemic is weak. We provide theoretical arguments that eigenvector centrality of a node approximately determines its risk to become infected.
... Therefore, the study of asymptomatic infection models has practical significance for the development of infectious disease dynamics [6]. This paper studies an extended SEIR epidemiological model called SEAIR [7], which has the potential in characterizing and controlling the evolution of the COVID-19 pandemic. SEAIR incorporates asymptomatic infected individuals into the classical SEIR model and also adds feedback vaccination control. ...
... Currently, the COVID-19 pandemic is still affecting countries over the world, and it has been extensively studied by numerous experts [2,6,7,10,16,19]. In fact, the occurrence and spread of infectious diseases is a dynamic process, which makes it imperative to dynamically control COVID-19 and rationally allocate limited resources. ...
... Based on Basnarkov[7], a SEAIR model is considered:S(t):The number of susceptible individuals at time t and S(t) ≥ 0. E(t): The number of exposed (or latent) individuals at time t and E(t) ≥ 0. A(t): The number of asymptomatic infected individuals at time t and A(t) ≥ 0. I(t): The number of infected individuals at time t and I(t) ≥ 0. R(t): The number of recovered individuals at time t and R(t) ≥ 0. ...
This paper studies a nonlinear epidemiological model called susceptible, exposed, asymptomatic infected, infected, and recovered (SEAIR) with a control input (vaccination), which helps to design a sliding mode vaccination strategy by tracking the prevalence of infectious diseases. First, the effective reproduction number is adjusted to the expected value to control infectious disease outbreaks, even with model parameter changes and systematic perturbations. Meanwhile, based on the sliding mode control theory, two control strategies, namely, first‐order sliding mode control and super‐twisting control, are proposed to control the spread of COVID‐19. Besides, an integral sliding surface associated with the effective reproduction number R0(t) is constructed to eliminate the chattering problem of the system. Moreover, the sliding mode controller is robust to uncertainties and external perturbations. The stability and finite‐time convergence of the closed‐loop control system are verified by Lyapunov's second method. Also, the effect of the two control strategies on the system performance is evaluated by numerical simulations. Furthermore, this paper presents a practical case simulation based on data from Wuhan. The results indicate that universal vaccination is crucial in the early stage of the COVID‐19 pandemic. As the epidemic progresses, vaccination is still an essential approach for epidemic prevention and control.
... Sun et al. looked at the SVIR model with both vaccination and incubation [13]. Basnarkov built the SEAIR model by capturing two dynamic characteristics of infectious diseases: delay and absence of symptoms [14]. A new SVEIS stochastic model was proposed based on the hypothesis that parameters satisfy Ornstein-Uhlenbeck process with mean regression [15]. ...
... and c 1 , c 2 are arbitrary positive constants. Additionally, if Eq. (14) holds, these functions are contractions. ...
Fabrizio and Caputo suggested an extraordinary definition of fractional derivative, which has been used in many fields. The SIDARTHE infectious disease model with regard to COVID-19 is studied by the new notion in this paper. Making use of the Banach fixed point theorem, the existence and uniqueness of the model’s solution are demonstrated. Then, an efficient method is utilized to deduce the iterative scheme. Finally, some numerical simulations of the model under various fractional orders and parameters are shown. From the computed result, we can see that it not only supports the theoretical demonstration, but also has an intensive insight into the characteristics of the model.
... Eigenvector centrality has been used as a suitable tool to study the spread of diseases in epidemics, as can be seen in Li et al. (2012) and Basnarkov (2021). ...
... In this specific study, betweenness centrality identified the potential students at greater risk. This is contrary to what is initially expected in disease transmission studies, as eigenvector centrality is generally the most appropriate method (Basnarkov 2021;Li et al. 2012). This is because the studied network is very large and disconnected, so the identification of vertices that are the point of communication between distant points of the graph is relevant. ...
With the advance of the COVID-19 pandemic, the emergence of new variants of the virus and peaks of the disease that occur seasonally until today, the study of disease proliferation becomes important. Thus, this study simulates the transmission of the virus in a network of undergraduate students from a Brazilian public university, which implemented the return to face-to-face classes at the beginning of 2022, using the concept of centrality in graphs. Several scenarios were considered, taking different groups as the first infected and analyzing the propagation effect of the disease in the network. The individuals who would represent the highest possible risk of inducing the disease, if infected, were detected through measures of centrality in networks. In addition, we also observed the peak of the disease, noting the highest number of infected people and the time to reach this peak, depending on the definition of the first infected. The identification of those first infected considering their importance in the network, via centrality measures, determines the disease cycle.
... Mathematical modeling is also studied in the COVID-19 epidemic by taking into account the effects of awareness programs in Nigeria [2]. Analysis of the Suspect-Latent-Asymptomatic-Infectious-Healed COVID-19 pattern of disease spread was studied, determining the threshold and the stability point [3]. Tian (2020) analyzed COVID-19 modeling based on morbidity data in Anhui, China [4]. ...
... The endemic stability point is obtained if the displacement of each subpopulation per day of t is constant, and infected people remain for t → . Based on Equation (1), it is assumed that the logistic growth is constant , obtained endemic point, It can be seen that 1 > 0, 2 > 0, 3 Table 1. Based on the Routh-Hurwitz criteria, 1 > 0, 2 > 0, 3 > 0 , and 1 2 − 3 > 0 , equivalently states that the characteristic roots of the polynomial system (6) all negative or R0 > 1 equivalent to the local asymptotically stable P1 endemic stability point. ...
This paper to analyzes the COVID-19 model with the growth of the logistics recruitment rate. Based on the model determined, the non-endemic stability points, threshold, and endemic stability points are obtained. The nonendemic stability point is asymptotically stable if the spread of COVID-19 decreases and vice versa. If the spread of COVID-19 increases, then the endemic stability P1 is globally asymptotically stable. Based on numerical simulations, the greater the recruitment rate, then the greater the number of susceptible and vaccinated subpopulation individuals. The smaller the value of the contact rate between infected individuals and those who are still healthy, the lower the number of infected individuals and vice versa, while the number of recovered subpopulation individuals is increasing. The greater the rate of treatment, the lower the number of infected individuals.
... The outbreak of infectious diseases has long posed significant threats to public health, social stability, and economic development. As such, accurately modeling epidemic transmission dynamics and devising practical prevention and containment strategies have become central challenges in public health research and management [1][2][3][4][5][6][7]. Global experiences in epidemic control have shown that the development of vaccines, the dissemination of disease-related information, and the spatial mobility of individuals all play crucial roles in shaping the trajectory of infectious disease spread. ...
While most existing epidemic models focus on the influence of isolated factors, infectious disease transmission is inherently shaped by the complex interplay of multiple interacting elements. To better capture real-world dynamics, it is essential to develop epidemic models that incorporate diverse, realistic factors. In this study, we propose a coupled disease-information spreading model on multiplex networks that simultaneously accounts for three critical dimensions: media influence, higher-order interactions, and population mobility. This integrated framework enables a systematic analysis of synergistic spreading mechanisms under practical constraints and facilitates the exploration of effective epidemic containment strategies. We employ a microscopic Markov chain approach (MMCA) to derive the coupled dynamical equations and identify epidemic thresholds, which are then validated through extensive Monte Carlo (MC) simulations. Our results show that both mass media dissemination and higher-order network structures contribute to suppressing disease transmission by enhancing public awareness. However, the containment effect of higher-order interactions weakens as the order of simplices increases. We also explore the influence of subpopulation characteristics, revealing that increasing inter-subpopulation connectivity in a connected metapopulation network leads to lower disease prevalence. Furthermore, guiding individuals to migrate toward less accessible or more isolated subpopulations is shown to effectively mitigate epidemic spread. These findings offer valuable insights for designing targeted and adaptive intervention strategies in complex epidemic settings.
... However, such frameworks inadequately capture multi-scale interactionse.g., cross-network delays between urban and rural regions or resource competition during outbreaks. Huang et al. 23 advanced this field by identifying influential links for targeted quarantine, and Basnarkov et al. 24 proposed a COVID-19-specific SEAIR model integrating asymptomatic carriers. Furthermore, network-augmented frameworks have extended the principles of fractional calculus and non-integer derivatives to systems with structured interactions [25][26][27][28] . ...
The co-evolution mechanisms between traffic mobility and disease transmission under resource constraints remain poorly understood. This study proposes a two-layer transportation network model integrating the Susceptible-Infectious-Susceptible (SIS) epidemic framework to address this gap. The model incorporates critical factors such as total medical resources, inter-network infection delays, travel willingness, and network topology. Through simulations, we demonstrate that increasing medical resources significantly reduces infection scale during outbreaks, while prolonging inter-network delays slows transmission rates but extends epidemic persistence. Complex network topologies amplify the impact of travel behavior on disease spread, and multi-factor interventions (e.g., combined resource allocation and delay extension) outperform single-factor controls in suppressing transmission. Furthermore, reducing network connectivity (lower average degree) proves effective in mitigating outbreaks, especially under low travel willingness. These findings highlight the necessity of coordinated policies that leverage resource optimization, travel regulation, and network simplification to manage epidemics. This work provides actionable insights for policymakers to design efficient epidemic control strategies in transportation-dependent societies.
... In order to simplify the model, the healing of both asymptomatic and infectious individuals is modeled with the same rate κ. Further details about this model can be found in [26]. The mathematical model under investigation is represented by the following system: ...
It is important to note that the process related to Covid-19 may exhibit random behavior due to environmental noise, and this factor should be taken into account. As a result, the modified Covid-19 model is evaluated using fractal-fractional derivatives in the sense of Caputo–Fabrizio, Caputo, and Atangana–Baleanu within a stochastic framework, aiming to create a more accurate representation of the Covid-19 outbreak. Mathematical analysis, including equilibrium points, the positivity of solutions, and the basic reproduction number for the deterministic model, is included in the study. The existence and uniqueness of solutions for the stochastic model are investigated under certain conditions. Additionally, the conditions for the existence of a global solution of the stochastic model are deduced, and the extinction of the infection within the model is studied. The outcomes of this model, incorporating memory effects, stochastic processes, and fractal properties, are supported by numerical simulations.
... Huang et al. [21] studied the identification of influential links in complex networks under the context of disease transmission, revealing that by eliminating influential links in the network, the number of infected individuals can be significantly reduced. Basnarkov et al. [22] proposed a Susceptible-Exposed-Asymptomatic-Infected-Recovered (SEAIR) epidemic transmission model for COVID-19 and derived a theoretical argument that the eigenvector centrality of a node approximately determines its infection risk. ...
The outbreak of diseases is influenced by various factors such as the total amount of resources and individual contacts. However, the co-evolution mechanism between individual travel behavior and disease transmission under limited resources remains unclear. In view of this, we construct a disease transmission model on a two-layer transportation network, considering the comprehensive effects of the total amount of medical resources, inter-network infection delay, travel willingness, and network topology. The simulation results show that increasing the total amount of resources can effectively reduce the disease scale in the transportation network during outbreaks. Additionally, an increase in inter-network infection delay can effectively slow down the disease transmission rate but prolongs the persistence of the disease in the population, affecting the regulation of infection scale by travel willingness. Meanwhile, the more complex the topology of the transportation network, the greater the impact of travel behavior on disease transmission. More importantly, compared to single-factor control, multi-factor combined control is more effective in inhibiting disease transmission. This paper provides new insights into the co-evolution mechanism of traffic travel behavior and disease transmission, and will offer valuable guidance for governments to control epidemic spread through transportation networks.
... In parallel, Ali et al. [13] and Awais [14] used fractional-order models to study COVID-19 dynamics in Pakistan. Similarly, Basnarkov [15] analyzed a SEAIR model, while Kumar et al. [16] proposed a modified SIR model with a non-monotonic incidence function for COVID-19 transmission in India. Other notable studies include [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][35][36][37]. ...
This study develops a modified SIR model (Susceptible–Infected–Recovered) to analyze the dynamics of the COVID-19 pandemic. In this model, infected individuals are categorized into the following two classes: Ia, representing asymptomatic individuals, and Is, representing symptomatic individuals. Moreover, accounting for the psychological impacts of COVID-19, the incidence function is nonlinear and expressed as Sg(Ia,Is)=βS(Ia+Is)1+α(Ia+Is). Additionally, the model is based on a symmetry hypothesis, according to which individuals within the same compartment share common characteristics, and an asymmetry hypothesis, which highlights the diversity of symptoms and the possibility that some individuals may remain asymptomatic after exposure. Subsequently, using the next-generation matrix method, we compute the threshold value (R0), which estimates contagiousness. We establish local stability through the Routh–Hurwitz criterion for both disease-free and endemic equilibria. Furthermore, we demonstrate global stability in these equilibria by employing the direct Lyapunov method and La-Salle’s invariance principle. The sensitivity index is calculated to assess the variation of R0 with respect to the key parameters of the model. Finally, numerical simulations are conducted to illustrate and validate the analytical findings.
... Within our application, we integrate four compartmental models to simulate the spread of COVID-19. These models include the Susceptible-Infectious-Recovered (SIR) [4], Susceptible-Exposed-Infectious-Recovered (SEIR) [5], Susceptible-Exposed-Asymptomatic-Infectious-Recovered (SEAIR) [13], and Susceptible-Exposed-Asymptomatic-Infectious-Quarantined-Hospitalized-Recovered-Deceased (SEAIQHRD) frameworks. ...
Infectious diseases present significant threats to humanity, with the COVID-19 pandemic wreaking havoc globally. Even though there are considerable educational resources to learn the fundamentals of epidemic modeling, a notable disparity exists between the relatively simple models employed for theoretical exposition and those utilized in practical applications to model COVID-19 dynamics. Comprehending these models demands substantial prerequisite knowledge and can be daunting for individuals new to epidemiological modeling and governing bodies. This study introduces COV19Sim Morocco, a web application designed to simulate COVID-19 dynamics using compartmental models. Through an interactive and user-friendly interface, the application presents a versatile tool for exploring COVID-19 data, simulating the spread of the virus, fitting models to available data, and generating predictions.
... For example, the SEIRUC model was considered in Tamilalagan et al. [11] to differentiate between asymptomatic and symptomatic individuals. In Basnarkov [12], the SEAIR model was considered, and stability analysis was performed, among others. As noted in the paper, there is an unknown variable that is solely dependent on time, which may not be practical in real-world situations [13,14]. ...
This paper presents a spatiotemporal reaction–diffusion model for epidemics to predict how the infection spreads in a given space. The model is based on a system of partial differential equations with the Neumann boundary conditions. First, we study the existence and uniqueness of the solution of the model using the semigroup theory and demonstrate the boundedness of solutions. Further, the proposed model's basic reproduction number is calculated using the eigenvalue problem. Moreover, the dynamic behavior of the disease‐free steady states of the model for R0<1 is investigated. The uniform persistence of the model is also discussed. In addition, the global asymptotic stability of the endemic steady state is examined. Finally, the numerical simulations validate the theoretical results.
... Therefore, our proposed model can be applied to the transmission of COVID-19. And for a more in-depth study, we can extend the susceptible-infected-removed (SIR) mathematical model into forms such as SEAIR (susceptible-exposed-asymptomatic-infectedrecovered) [55,56] according to its properties such as the presence of asymptomatic infection and incubation period. ...
The process of information diffusion can cause changes in the emotions of individuals, which in turn can have an impact on the process of disease spreading. This paper describes a coupled information-disease spreading model based on multiplex networks, taking into account individual emotions, which can be used to investigate the interaction between positive and negative information diffusion, emotion accumulation, and disease spreading. In our model, emotions act as a mediator of the interaction between information diffusion and disease spreading, with individuals reacting emotionally based on the information they receive, and emotions in turn influencing disease infection and recovery through accumulation. In addition, to measure the impact of emotions, we introduce two indicators: the emotion threshold and the immune impact factor. The Micro-Markov chain approach is used for the theoretical analysis to derive the dynamic evolution process of the model and to calculate the prevalence threshold. We validate the theoretical analysis through extensive numerical simulations. The results show that individual emotions can have an impact on the disease spreading process. Specifically, positive emotions can effectively inhibit the spread of disease, slow down the rate of an outbreak, and reduce outbreak size in a population, while negative emotions can greatly facilitate the spread of disease, accelerate the rate of disease prevalence, and increase outbreak size.
... Herein is part of the novelty of the presented work, as we further develop a mathematical framework that permits non-exponential and non-Erlang distributed durations of infection while retaining model formulation as a system of ODEs. While such a framework has been developed under the context of SIS and SIR models [18,19], it has yet to be cast into more elaborate compartmental models such as a SEAIR analog [20]. ...
Sexually transmitted diseases (STDs) are detrimental to the health and economic well-being of society. Consequently, predicting outbreaks and identifying effective disease interventions through epidemiological tools, such as compartmental models, is of the utmost importance. Unfortunately, the ordinary differential equation compartmental models attributed to the work of Kermack and McKendrick require a duration of infection that follows the exponential or Erlang distribution, despite the biological invalidity of such assumptions. As these assumptions negatively impact the quality of predictions, alternative approaches are required that capture how the variability in the duration of infection affects the trajectory of disease and the evaluation of disease interventions. So, we apply a new family of ordinary differential equation compartmental models based on the quantity person-days of infection to predict the trajectory of disease. Importantly, this new family of models features non-exponential and non-Erlang duration of infection distributions without requiring more complex integral and integrodifferential equation compartmental model formulations. As proof of concept, we calibrate our model to recent trends of chlamydia incidence in the U.S. and utilize a novel duration of infection distribution that features periodic hazard rates. We then evaluate how increasing STD screening rates alter predictions of incidence and disability adjusted life-years over a five-year horizon. Our findings illustrate that our family of compartmental models provides a better fit to chlamydia incidence trends than traditional compartmental models, based on Akaike information criterion. They also show new asymptomatic and symptomatic infections of chlamydia peak over drastically different time frames and that increasing the annual STD screening rates from 35% to 40%-70% would annually avert 6.1-40.3 incidence while saving 1.68-11.14 disability adjusted life-years per 1000 people. This suggests increasing the STD screening rate in the U.S. would greatly aid in ongoing public health efforts to curtail the rising trends in preventable STDs.
... Since then, various extensions of this mathematical model have been proposed that include more detailed information, such as spatial heterogeneity, seasonality, age-structures, etc., and significant progress has been made in almost all of these aspects. Most of these mathematical epidemic models assume the spread of disease by random contact between individuals, which is usually formulated by the law of group action [3,4], so behavioral effects are excluded. However, in addition to the disease itself, the mechanisms of disease transmission possibly involve social, economic and psychological factors. ...
To investigate the influence of human behavior on the spread of COVID-19, we propose a reaction–diffusion model that incorporates contact rate functions related to human behavior. The basic reproduction number R0 is derived and a threshold-type result on its global dynamics in terms of R0 is established. More precisely, we show that the disease-free equilibrium is globally asymptotically stable if R0≤1; while there exists a positive stationary solution and the disease is uniformly persistent if R0>1. By the numerical simulations of the analytic results, we find that human behavior changes may lower infection levels and reduce the number of exposed and infected humans.
... Law K B et al. [14] modified the SIR model to specifically simulate the early depleting transmission dynamics of COVID-19 to better predict its temporal trend in Malaysia. Basnarkov L et al. [15] proposed an SEAIR model for COVID-19 transmission simulation. The model was theoretically analyzed in continuoustime compartmental version and discrete-time version on random regular graphs and complex networks. ...
In the classical infectious disease compartment model, the parameters are fixed. In reality, the probability of virus transmission in the process of disease transmission depends on the concentration of virus in the environment, and the concentration depends on the proportion of patients in the environment. Therefore, the probability of virus transmission changes with time. Then how to fit the parameters and get the trend of the parameters changing with time is the key to predict the disease course with the model. In this paper, based on the US COVID-19 epidemic statistics during calibration period, the parameters such as infection rate and recovery rate are fitted by using the linear regression algorithm of machine science, and the laws of these parameters changing with time are obtained. Then a SIR model with time delay and vaccination is proposed, and the optimal control strategy of epidemic situation is analyzed by using the optimal control theory and Pontryagin maximum principle, which proves the effectiveness of the control strategy in restraining the transmission of COVID-19. The numerical simulation results show that the time-varying law of the number of active cases obtained by our model basically conforms to the real changing law of the US COVID-19 epidemic statistics during calibration period. In addition, we have predicted the changes in the number of active cases in the COVID-19 epidemic in the USA over time in the future beyond the calibration cycle, and the predicted results are more in line with the actual epidemic data.
... Since then, various extensions of this mathematical model have been proposed that include more detailed information, such as spatial heterogeneity, seasonality, age-structures, etc., and significant progress has been made in almost all of these aspects. Most of these mathematical epidemic models assume the spread of disease by random contact between individuals, which is usually formulated by the law of group action [3,4], so behavioral effects are excluded. However, in addition to the disease itself, the mechanisms of disease transmission possibly involve social, economic and psychological factors. ...
COVID-19 is a highly infectious disease spreading through human droplets and contact. To investigate the effect of human behavior changes on the spread of COVID-19, a reaction-diffusion model that contains contact rate functions related to human behavior is studied. The basic reproduction number for this system is derived and a threshold-type result on its global dynamics in terms of is established in this paper. More precisely, we show that the disease-free equilibrium is globally asymptotically stableif and the system admits a positive solution and the disease is uniformly persistent if . By the numerical simulations of the analytic results, we find that human behavior changes may lower infection levels and reduce the number of exposed and infected humans.
... SIR divides the population into the three SIR compartments. Various models have been derived and developed, based on the SIR model, including the Susceptible-Exposed-Infected-Removed (SEIR) [22,23] and the Susceptible-Exposed-Asymptomatic-Infectious-Recovered (SEAIR) model [24] . Mathematical modeling based on numerical solutions of systems of Ordinary Differential Equations (ODEs) has also been used to study COVID-19 spread. ...
A physics-informed neural network (PINN) embedded with the susceptible-infected-removed (SIR) model is devised to understand the temporal evolution dynamics of infectious diseases. Firstly, the effectiveness of this approach is demonstrated on synthetic data as generated from the numerical solution of the susceptible-asymptomatic-infected-recovered-dead (SAIRD) model. Then, the method is applied to COVID-19 data reported for Germany and shows that it can accurately identify and predict virus spread trends. The results indicate that an incomplete physics-informed model can approach more complicated dynamics efficiently. Thus, the present work demonstrates the high potential of using machine learning methods, e.g., PINNs, to study and predict epidemic dynamics in combination with compartmental models.
... In this paper, we consider a compartmental model with five compartments, in which, at any time, an individual can be either susceptible (S), exposed to the disease but still unable to spread it (E), asymptomatic infectious (A), symptomatic infectious (I), or recovered/removed (R). We refer to this compartmental model as the SEAIR model [42,43]. A common approach is to consider a Poisson process framework, in which events, such as infections and spontaneous transitions, can occur at any time with a certain rate. ...
Epidemic models are crucial to understand how an infectious disease spreads in a population and to devise the best containment strategies. Compartmental models can provide a fine-grained description of the evolution of an epidemic when microscopic information on the network of contacts among individuals is available. However, coarser-grained descriptions prove also to be useful in many aspects. They allow to derive closed expressions for key parameters, such as the basic reproduction number, to understand the relationship between the model parameters, and also to derive fast and reliable predictions of macroscopic observables for a disease outbreak. The so-called population models can be developed at different levels of coarse-graining, so it is crucial to determine: (i) to which extent and how the existing correlations in the contact network have to be included in these models and (ii) what is their impact on the model ability to reproduce and predict the time evolution of the populations at the various stage of the disease. In this work, we address these questions through a systematic analysis of two discrete-time SEAIR (susceptible-exposed-asymptomatic-infected-recovered) population models: the first one developed assuming statistical independence at the level of individuals, and the other one assuming independence at the level of pairs. We provide a detailed derivation and analysis of both models, focusing on their capability to reproduce an epidemic process on different synthetic networks, and then comparing their predictions under scenarios of increasing complexity. We find that, although both models can fit the time evolution of the compartment populations obtained through microscopic simulations, the epidemic parameters adopted by the individual-based model for this purpose may significantly differ from those of the microscopic simulations. However, pair-based model provides not only more reliable predictions of the dynamical evolution of the variables but also a good estimation of the epidemic parameters. The difference between the two models is even more evident in the particularly challenging scenario when one or more variables are not measurable, and therefore are not available for model tuning. This occurs for instance with asymptomatic infectious individuals in the case of COVID-19, an issue that has become extremely relevant during the recent pandemic. Under these conditions, the pairwise model again proves to perform much better than the individual-based representation, provided that it is fed with adequate information which, for instance, to be collected, may require a more detailed contact tracing. Overall, our results thus hallmark the importance of acquiring the proper empirical data to fully unfold the potentialities of models incorporating more sophisticated assumptions on the correlations among nodes in the contact network.
... Pandemic analysis models have been frequently used to assess the risks of the Covid-19 spread. Common methods use the parametric models to estimate the number of patients for each stage of the illness [1][2][3]. Prediction of the state of the pandemic is done through aggregating the statistics and using those statistics in models. These models are used to track the state of the Covid-19 pandemic for a short time [4]. ...
Current state of art approaches such as the susceptible-infected-removed model and machine learning models are not optimized for modeling the risks of individuals and modeling the effects of local restrictions. To improve the drawback of these approaches, the feedback processing framework is proposed where previously accumulated global statistics and the model estimates generated from the spatial-temporal data are combined to improve the performance of the local prediction. The proposed framework is evaluated in three processing stages: generation of the simulation dataset, feedback analysis, and evaluation for the spatial-temporal and real-time pandemic analysis. In the data generation stage, the corresponding state of the illness for each person is modeled by a Markov stochastic process. In this stage, the parameters such as the reproduction rate, symptomatic rate, asymptomatic rate, population count, infected count, and the average mobility rate are used to update the individual's Covid-19 status and the individual's movements. The movement data of each person is generated randomly for several places of interest. In the feedback analysis stage, both the aggregated statistics and the local event data are combined in a linear model to infer a score for the Covid-19 probability of the person. In this respect, a stochastic model can be used to approximate the local statistics. In the evaluation stage, the result of the feedback analysis for all the interactions is used to classify the state of the individuals periodically. Later the accuracy of the evaluation for each person is obtained by comparing the individual's prediction with the real data generated in the same time interval. The Kappa scores independent from different populations, locations, and mobility rates obtained for every interaction indicate a significant difference from the random statistics.
... With the emergence and outbreak of COVID-19 [3,30] in recent years, infectious disease models have become one of the most popular research topics. To study the spread and dynamics of COVID-19, most scholars use the SIR (susceptible-infected-recovered) [3,28], SEIR (susceptibleexposed-infected-recovered) [25,30] and SEAIR (susceptible-exposed-asymptomatic-infectiousremoved) [2,46] models to describe the spread of COVID-19. Meanwhile, the classical SIS model has received great attention in mathematical epidemiology. ...
To study the influence of the moving front of the infected interval and the spatial movement of individuals on the spreading or vanishing of infectious disease, we consider a nonlocal SIS (susceptible-infected-susceptible) reaction-diffusion model with media coverage, hospital bed numbers and free boundaries. The principal eigenvalue of the integral operator is defined, and the impacts of the diffusion rate of infected individuals and interval length on the principal eigenvalue are analyzed. Furthermore, the sufficient conditions for spreading and vanishing of the disease are derived. Our results show that large media coverage and hospital bed numbers are beneficial to the prevention and control of disease. The difference between the model with nonlocal diffusion and that with local diffusion is also discussed and nonlocal diffusion leads more possibilities.
... Such a model and its extensions have been widely used to model the COVID-19 epidemic since a significant proportion of people who contracted CoVID-19 had no symptoms (De la Sen et al., 2020;Jia and Chen, 2021;Basnarkov, 2021). It was also used to study other diseases like Influenza A (H1N1) (Jin et al., 2011). ...
This thesis focuses on the sensitivity analysis of stochastic models. These models include uncertainties that originate mainly from two sources: the parametric uncertainty due to the lack of knowledge of parameters and the intrinsic randomness that represents the noise inherent to the model coming from the way chance intervenes in the description of the modeled phenomenon. The presence of intrinsic randomness is a challenge in sensitivity analysis because, on the one hand, it is generally hidden and therefore cannot be characterized and, on the other hand, it acts as noise when evaluating the impact of the parameters on the model output However, in epidemiology, the issues associated with the sensitivity of a model can be important in the control of epidemics because they impact the decisions made on the basis of this model. This thesis studies approaches for sensitivity analysis of stochastic models such as epidemiological models based on stochastic processes, in the framework of variance-based analysis. In a general context, we introduce a method for estimating sensitivity indices that optimizes the trade-off between the number of input parameter values of the model and the number of replications of model evaluation in each of these values. For this method, we consider the class of quantities of interest of stochastic model outputs that are in the form of conditional expectations with respect to uncertain parameters. In the context of estimation of sensitivity indices by the Monte Carlo method, we control the quadratic risk of the estimators, show its convergence and find a trade-off between the bias related to the presence of the intrinsic randomness and the variance. In the specific context of stochastic compartmental models in epidemiology, we characterize the intrinsic randomness of the stochastic processes on which these models are based. These stochastic processes can be Markovian or non-Markovian. For Markovian processes, we use Gillespie algorithms to make explicit the intrinsic randomness and to separate it from uncertain parameters. Regarding non-Markovian processes, we extend to a large class of compartmental models the Sellke construction, which was originally introduced to describe epidemic dynamics of the SIR model in a framework that is not necessarily Markovian. This extension has allowed us to develop an algorithm that generates exact trajectories in a non-Markovian framework for a large class of compartmental models but also to be able to separate intrinsic randomness from parameter uncertainty in the output of these models. Thus, for both types of processes, Markovian and non-Markovian, the separation of the two sources of uncertainty has been obtained and it allows to represent model outputs as a deterministic function of the uncertain parameters and the variables representing the intrinsic randomness. When the uncertainty on the parameters is assumed to be independent of the intrinsic randomness, this representation allows to assess the contributions of the intrinsic randomness on the model outputs, in addition to the contributions of the parameters. It is also possible to characterize different interactions. This thesis has contributed to develop an approach to estimate sensitivity indices and to evaluate the contribution of intrinsic randomness in compartmental models in epidemiology based on stochastic processes.
... Wood et al. [65] investigated the effectiveness of increasing healthcare capacity and extending the period of isolation. Some studies distinguish between and incorporate both asymptomatic and symptomatic persons, who play an important role in the COVID-19 pandemic (e.g., [3,9,19,20,25,29,40,54,62]). Moreover, the infectiousness of asymptomatic infected cases has been reported to be lower than that of symptomatically infected cases [22,39]. ...
We provided a framework of a mathematical epidemic modeling and a countermeasure against the novel coronavirus disease (COVID-19) under no vaccines and specific medicines. The fact that even asymptomatic cases are infectious plays an important role for disease transmission and control. Some patients recover without developing the disease; therefore, the actual number of infected persons is expected to be greater than the number of confirmed cases of infection. Our study distinguished between cases of confirmed infection and infected persons in public places to investigate the effect of isolation. An epidemic model was established by utilizing a modified extended Susceptible-Exposed-Infectious-Recovered model incorporating three types of infectious and isolated compartments, abbreviated as SEIIIHHHR. Assuming that the intensity of behavioral restrictions can be controlled and be divided into multiple levels, we proposed the feedback controller approach to implement behavioral restrictions based on the active number of hospitalized persons. Numerical simulations were conducted using different detection rates and symptomatic ratios of infected persons. We investigated the appropriate timing for changing the degree of behavioral restrictions and confirmed that early initiating behavioral restrictions is a reasonable measure to reduce the burden on the health care system. We also examined the trade-off between reducing the cumulative number of deaths by the COVID-19 and saving the cost to prevent the spread of the virus. We concluded that a bang-bang control of the behavioral restriction can reduce the socio-economic cost, while a control of the restrictions with multiple levels can reduce the cumulative number of deaths by infection.
... Data structuring can prevent dichotomization-and cutoff-related errors (Royston et al., 2006;Marquand et al., 2016). Spatial-temporal data patterns also distinguish non-binary conditions -such as COVID-19, an infection associated with more than two clinical presentations (Basnarkov, 2021). The validated and clinically applicable examples described here follow a classic prescription for method development: 'the goal is to discover things we neither knew nor expected, and to see relationships and connections among the elements, whether previously suspected or not … this process is not driven by hypothesis and should be as model-independent as possible' (Brown and Botstein, 1999). ...
Topics expected to influence personalized medicine (PM), where medical decisions, practices, and treatments are tailored to the individual patient, are reviewed. Lack of discrimination due to different biological conditions that express similar values of numerical variables (ambiguity) is regarded to be a major potential barrier for PM. This material explores possible causes and sources of ambiguity and offers suggestions for mitigating the impacts of uncertainties.
Three causes of ambiguity are identified: (1) delayed adoption of innovations, (2) inadequate emphases, and (3) inadequate processes used when new medical practices are developed and validated. One example of the first problem is the relative lack of medical research on “compositional data” –the type that characterizes leukocyte data. This omission results in erroneous use of data abundantly utilized in medicine, such as the blood cell differential. Emphasis on data output ‒not biomedical interpretation that facilitates the use of clinical data‒ exemplifies the second type of problems. Reliance on tools generated in other fields (but not validated within biomedical contexts) describes the last limitation.
Because reductionism is associated with these problems, non-reductionist alternatives are reviewed as potential remedies. Data structuring (converting data into information) is considered a key element that may promote PM. To illustrate a process that includes data-information-knowledge and decision-making, previously published data on COVID-19 are utilized.
It is suggested that ambiguity may be prevented or ameliorated. Provided that validations are grounded on biomedical knowledge, approaches that describe certain criteria – such as non-overlapping data intervals of patients that experience different outcomes, immunologically interpretable data, and distinct graphic patterns – can inform, at personalized bases, earlier and/or with fewer observations.
... Law K B et al. [14] modified the SIR model to specifically simulate the early depleting transmission dynamics of COVID-19 to better predict its temporal trend in Malaysia. Basnarkov L et al. [15] proposed an SEAIR model for COVID-19 spread simulation. The model was theoretically analyzed in continuous-time compartmental version and discrete-time version on random regular graphs and complex networks. ...
In the classical infectious disease compartment model, the parameters are fixed. In reality, the probability of virus transmission in the process of disease transmission depends on the concentration of virus in the environment, and the concentration depends on the proportion of patients in the environment. Therefore, the probability of virus transmission changes with time. Then how to fit the parameters and get the trend of the parameters changing with time is the key to predict the disease course with the model. In this paper, based on the US COVID-19 epidemic statistics during calibration period, the parameters such as incidence rate and recovery rate are fitted by using the linear regression algorithm of machine science, and the laws of these parameters changing with time are obtained. Then a SIR model with time delay and vaccination is proposed, and the optimal control strategy of epidemic situation is analyzed by using the optimal control theory and Pontryagin maximum principle, which proves the effectiveness of the control strategy in restraining the spread of COVID-19. The numerical simulation results show that the time-varying law of the number of active cases obtained by our model basically conforms to the real changing law of the US COVID-19 epidemic statistics during calibration period. In addition, we have predicted the changes of the number of active cases in the COVID-19 epidemic in the United States over time in the future beyond the calibration cycle, and the predicted results are more in line with the actual epidemic data.
... Some authors have adapted the classical SIR structure; see, e.g., Cooper et al. (2020), Nguemdjo et al. (2020). Others have extended the model to include a latent or exposed compartment, such as Chen et al. (2020), Yang and Wang (2020), while others have gone further to include both latent/exposed and asymptomatic compartments, as in , , Basnarkov (2021), Li et al. (May 2020), Rȃdulescu et al. (2020), Tsay et al. (2020). We refer to the general class of models that extend the classical SIR model structure to include latent/exposed (L) and asymptomatic (A) compartments as SLIAR-type models. ...
Various vaccines have been approved for use to combat COVID-19 that offer imperfect immunity and could furthermore wane over time. We analyze the effect of vaccination in an SLIARS model with demography by adding a compartment for vaccinated individuals and considering disease-induced death, imperfect and waning vaccination protection as well as waning infections-acquired immunity. When analyzed as systems of ordinary differential equations, the model is proven to admit a backward bifurcation. A continuous time Markov chain (CTMC) version of the model is simulated numerically and compared to the results of branching process approximations. While the CTMC model detects the presence of the backward bifurcation, the branching process approximation does not. The special case of an SVIRS model is shown to have the same properties.
... The spectrum of mathematical models applied for the COVID-19 pandemic ranges from the simplest SIR to rather complex SIDARTHE [1][2][3][4][5][6][7], which are used for assessment of different aspects of the epidemics. One of the major features of these models is their Markovian nature, which considers transitions from one state to another to be independent on the past. ...
We introduce non-Markovian SIR epidemic spreading model inspired by the characteristics of the COVID-19, by considering discrete- and continuous-time versions. The distributions of infection intensity and recovery period may take an arbitrary form. By taking corresponding choice of these functions, it is shown that the model reduces to the classical Markovian case. The epidemic threshold is analytically determined for arbitrary functions of infectivity and recovery and verified numerically. The relevance of the model is shown by modeling the first wave of the epidemic in Italy, Spain and the UK, in the spring, 2020.
... The D-SUCR model allows us to easily evaluate the actual pandemic situation in all stages of multiple waves in a region or country, using an evolutionary computation-based system identification algorithm under rational epidemiological constraints. To our knowledge, most previously studied epidemic models can only describe the evolution of a first or separate wave [14,45,2,17,39], and with the emergence of mutant viruses, this is no longer adequate to control the pandemic. Our proposed D-SUCR model is able to dynamically provide accurate descriptions over multiple waves of the COVID-19 pandemic, which will provide more accurate and effective scientific guidance to policy decision-makers. ...
The novel coronavirus disease 2019 (COVID-19), caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has unique epidemiological characteristics that include presymptomatic and asymptomatic infections, resulting in a large proportion of infected cases being unconfirmed, including patients with clinical symptoms who have not been identified by screening. These unconfirmed infected individuals move and spread the virus freely, presenting difficult challenges to the control of the pandemic. To reveal the actual pandemic situation in a given region, a simple dynamic susceptible-unconfirmed-confirmed-removed (D-SUCR) model is developed taking into account the influence of unconfirmed cases, the testing capacity, the multiple waves of the pandemic, and the use of non-pharmaceutical interventions. Using this model, the total numbers of infected cases in 51 regions of the USA and 116 countries worldwide are estimated, and the results indicate that only about 40% of the true number of infections have been confirmed. In addition, it is found that if local authorities could enhance their testing capacities and implement a timely strict quarantine strategy after identifying the first infection case, the total number of infected cases could be reduced by more than 90%. Delay in implementing quarantine measures would drastically reduce their effectiveness.
... As a result of the study, the proposed SIR model showed excellent performance and provided information related to the spread of the virus over time that could not be obtained with data alone. Basnarkov [19] proposed a Susceptible-Exposed-Asymptomatic-Infectious-Recovered (SEAIR) model. The proposed SEAIR is a model that adds a symptom compartment to the SEIR model, and the infection characteristics of COVID-19 were analyzed. ...
Following the outbreak of the COVID-19 pandemic, the continued emergence of major variant viruses has caused enormous damage worldwide by generating social and economic ripple effects, and the importance of PHSMs (Public Health and Social Measures) is being highlighted to cope with this severe situation. Accordingly, there has also been an increase in research related to a decision support system based on simulation approaches used as a basis for PHSMs. However, previous studies showed limitations impeding utilization as a decision support system for policy establishment and implementation, such as the failure to reflect changes in the effectiveness of PHSMs and the restriction to short-term forecasts. Therefore, this study proposes an LSTM-Autoencoder-based decision support system for establishing and implementing PHSMs. To overcome the limitations of existing studies, the proposed decision support system used a methodology for predicting the number of daily confirmed cases over multiple periods based on multiple output strategies and a methodology for rapidly identifying varies in policy effects based on anomaly detection. It was confirmed that the proposed decision support system demonstrated excellent performance compared to models used for time series analysis such as statistical models and deep learning models. In addition, we endeavored to increase the usability of the proposed decision support system by suggesting a transfer learning-based methodology that can efficiently reflect variations in policy effects. Finally, the decision support system proposed in this study provides a methodology that provides multi-period forecasts, identifying variations in policy effects, and efficiently reflects the effects of variation policies. It was intended to provide reasonable and realistic information for the establishment and implementation of PHSMs and, through this, to yield information expected to be highly useful, which had not been provided in the decision support systems presented in previous studies.
... The SEAIR epidemic spreading model has been proposed in [50], but modification in that model is given here. Let be susceptible individuals, be the exposed individuals, be infected individuals, be quarantined individuals, and be recovery people. ...
This contribution proposes a numerical scheme for solving fractional parabolic partial differential equations (PDEs). One of the advantages of using the proposed scheme is its applicability for fractional and integer order derivatives. The scheme can be useful to get conditions for obtaining a positive solution to epidemic disease models. A COVID-19 mathematical model is constructed, and linear local stability conditions for the model are obtained; afterward, a fractional diffusive epidemic model is constructed. The numerical scheme is constructed by employing the fractional Taylor series approach. The proposed fractional scheme is second-order accurate in space and time and unconditionally stable for parabolic PDEs. In addition to this, convergence conditions are obtained by employing a proposed numerical scheme for the fractional differential equation of susceptible individuals. The scheme is also compared with existing numerical schemes, including the non-standard finite difference method. From theoretical analysis and graphical illustration, it is found that the proposed scheme is more accurate than the so-called existing non-standard finite difference method, which is a method with notably good boundedness and positivity properties.
... Mathematical models of ordinary differential equations have significance in studying the dynamic behavior of infectious diseases. In recent times, many mathematical models including mumps virus [10], ebola virus disease [11], dengue fever disease [12], rubella disease [13], influenza transmission [14], zika virus transmission [15], COVID-19 pandemic [16][17][18] and many others have been formulated using differential equations. ...
The human immunodeficiency virus (HIV) mainly attacks CD4⁺ T cells in the host. Chronic HIV infection gradually depletes the CD4⁺ T cell pool, compromising the host’s immunological reaction to invasive infections and ultimately leading to acquired immunodeficiency syndrome (AIDS). The goal of this study is not to provide a qualitative description of the rich dynamic characteristics of the HIV infection model of CD4⁺ T cells, but to produce accurate analytical solutions to the model using the modified iterative approach. In this research, a new efficient method using the new iterative method (NIM), the coupling of the standard NIM and Laplace transform, called the modified new iterative method (MNIM), has been introduced to resolve the HIV infection model as a class of system of ordinary differential equations (ODEs). A nonlinear HIV infection dynamics model is adopted as an instance to elucidate the identification process and the solution process of MNIM, only two iterations lead to ideal results. In addition, the model has also been solved using NIM and the fourth order Runge–Kutta (RK4) method. The results indicate that the solutions by MNIM match with those of RK4 method to a minimum of eight decimal places, whereas NIM solutions are not accurate enough. Numerical comparisons between the MNIM, NIM, the classical RK4 and other methods reveal that the modified technique has potential as a tool for the nonlinear systems of ODEs.
... While The last expression indicates that the Infectiousness probability vector is eigenvector of scaled connectivity matrix in weak epidemic. This is similar to the previous result that the principal eigenvector determines the probabilities of infection in SEAIR model [32]. ...
In the light of several major epidemic events that emerged in the past two decades, and emphasized by the COVID-19 pandemics, the non-Markovian spreading models occurring on complex networks gained significant attention from the scientific community. Following this interest, in this article, we explore the relations that exist between the mean-field approximated non-Markovian SEIS (Susceptible–Exposed–Infectious–Susceptible) and the classical Markovian SIS, as basic reoccurring virus spreading models in complex networks. We investigate the similarities and seek for equivalences both for the discrete-time and the continuous-time forms. First, we formally introduce the continuous-time non-Markovian SEIS model, and derive the epidemic threshold in a strict mathematical procedure. Then we present the main result of the paper that, providing certain relations between process parameters hold, the stationary-state solutions of the status probabilities in the non-Markovian SEIS may be found from the stationary state probabilities of the Markovian SIS model. This result has a two-fold significance. First, it simplifies the computational complexity of the non-Markovian model in practical applications, where only the stationary distributions of the state probabilities are required. Next, it defines the epidemic threshold of the non-Markovian SEIS model, without the necessity of a thrall mathematical analysis. We present this result both in analytical form, and confirm the result trough numerical simulations. Furthermore, as of secondary importance, in an analytical procedure we show that each Markovian SIS may be represented as non-Markovian SEIS model.
... Basnarkov [36] studied SEAIR model using two different approaches, first one is based on differential equations and other is based on discrete-time epidemic model. In recent years, intuitionistic fuzzy sets have been preferred in the fuzzy modeling. ...
The outbreak of COVID-19 has become a global pandemic as announced by World Health Organisation. As India has already met the two waves, named first and second wave, it is assumed that COVID-19 will again strike in India in the form of third wave. The peak during the upcoming third wave and determination of the approximated maximum number of COVID-19 infected cases and deaths at a particular day becomes crucial for India. To determine the peak of infectious curve, this article proposed a hybrid fuzzy time series forecasting model based on particle swarm optimization and fuzzy c-mean technique, named as fuzzy time series particle swarm optimization extended fuzzy c-mean technique. The proposed model works in two phases. In phase-I, particle swarm optimization extended fuzzy c-mean method is used to form initial intervals with the help of centroids, while in phase-II, these intervals are updated to form subintervals. In the present article, a fitness function is developed for particle swarm optimization to increase its convergence speed and basic fuzzy c-mean is extended by using an exponential function to tolerate the effect of outliers, named as extended fuzzy c-mean technique. The effectiveness of the proposed model has been tested based on mean square error and root mean square error on first and second wave COVID-19 data, and the obtained results are very close to the existing data of COVID-19 with less error rate. Thus, the proposed model is suitable to forecast a better approximation value of COVID-19 infected cases and deaths in India during the upcoming third wave. This study demonstrates that third wave of COVID-19 could occur in India, while also illustrating that it is unlikely for any such resurgence to be as large as the second wave. The proposed model predicts that the peak of third wave will occur approximately after 40–70 days from the mid of December. Furthermore, the impact of vaccination on infected cases and deaths during the upcoming third wave in India is also studied. With the implementation of the vaccine on the Indian people, the peak of COVID-19 infected during third wave will be shifted in forward direction. On the basis of the proposed model, government authorities will be enabling to know expected required resources such as hospital patient beds, ICU beds, and oxygen concentrators during the upcoming outspread of COVID-19 like disease in future.
Traditional infectious disease models often use fixed compartments to represent different states of individuals. However, these models can be limited in accurately reflecting real-world conditions of individuals. In this study, we integrate quantum mechanics into infectious disease modeling, developing a quantum mechanics-based model that effectively addresses the limitations of traditional compartmental models and introduces a novel approach to understanding disease dynamics. Firstly, we examined the individual infection process and the model’s evolutionary dynamics, deriving both the disease-free equilibrium point and the model's basic reproduction number. Secondly, the proposed model is simulated on a quantum circuit. The simulation results are utilized to analyze the model’s parameter sensitivity and verify its rationality. The results indicate that the model’s predictions align with the general patterns of viral transmission and are capable of replicating the structural attributes of compartmental models. Finally, we apply the model to simulate the spread of COVID-19. The observed similarity between the simulated results and actual infection trends demonstrates the model's effectiveness in accurately capturing viral transmission dynamics. Comparative experiments show that the proposed model significantly improves accuracy over traditional models. By leveraging quantum mechanics, our method offers a fresh perspective in infectious disease modeling, broadening the application of quantum mechanics methodologies in understanding information propagation within the macroscopic world.
The frequent occurrence of major public health emergencies (MPHEs) significantly challenges national security, economic stability, social operation and the safety of people's lives and property worldwide. Consequently, enhancing the emergency management of MPHEs is critically urgent. This paper constructs a game model involving local government, social organisations, and the public for MPHE management, exploring strategy combinations and influencing factors across various scenarios. Several results were obtained. (1) Local government, social organisations, and the public each have positive and negative strategy choices based on cost–benefit analysis, leading to eight different strategy combinations. Furthermore, all three take positive strategies as the optimal way to achieve the game equilibrium. (2) The transformation of strategy combinations is primarily influenced by the cost-benefit gap and the strategic decisions of local government. (3) Altering a subject's initial strategy value doesn't change its final choice but impacts the time to achieve a stable strategy equilibrium. The severity of local government punishments on social organisations influences their strategic choices and the time to optimal strategy, whereas rewards to the public or social organisations only affect the time to achieve this strategy. The findings of this study can not only help improve the collaborative governance system of MPHEs but also provide scientific guidance on how governments can manage MPHEs.
The complex networks exhibit significant heterogeneity in node connections, resulting in a few nodes playing critical roles in various scenarios, including decision-making, disease control, and population immunity. Therefore, accurately identifying these influential nodes that play crucial roles in networks is very important. Many methods have been proposed in different fields to solve this issue. This paper focuses on the different types of disassortativity existing in networks and innovatively introduces the concept of disassortativity of the node, namely, the inconsistency between the degree of a node and the degrees of its neighboring nodes, and proposes a measure of disassortativity of the node (DoN) by a step function. Furthermore, the paper analyzes and indicates that in many real-world network applications, such as online social networks, the influence of nodes within the network is often associated with disassortativity of the node and the community boundary structure of the network. Thus, the influential metric of node based on disassortativity and community structure (mDC) is proposed. Extensive experiments are conducted in synthetic and real networks, and the performance of the DoN and mDC is validated through network robustness experiments and immune experiment of disease infection. Experimental and analytical results demonstrate that compared to other state-of-the-art centrality measures, the proposed methods (DoN and mDC) exhibits superior identification performance and efficiency, particularly in non-disassortative networks and networks with clear community structures. Furthermore, we find that the DoN and mDC exhibit high stability to network noise and inaccuracies of the network data.
Vaccination and social media play pivotal roles in affecting disease transmission. Research has shown that disease transmission can be subject to random events. Consequently, we develop a stochastic infectious disease dynamical model that incorporates saturated media coverage and vaccination age. The Itô’s formula and the Lyapunov function method are applied to study the extinction behavior of the disease and the existence of a unique ergodic stationary distribution. The findings suggest that the media effect is delayed and cannot eliminate the disease completely. To directly control disease transmission, a combination of high-intensity noise disturbance and low vaccine wane rate is required. Furthermore, the shorter the disease incubation period, the more difficult it is to control.
To study the influence of the moving front of the infected interval and the spatial movement of individuals on the spreading or vanishing of infectious disease, we consider a nonlocal susceptible–infected–susceptible (SIS) reaction–diffusion model with media coverage, hospital bed numbers and free boundaries. The principal eigenvalue of the integral operator is defined, and the impacts of the diffusion rate of infected individuals and interval length on the principal eigenvalue are analyzed. Furthermore, sufficient conditions for spreading and vanishing of the disease are derived. Our results show that large media coverage and hospital bed numbers are beneficial to the prevention and control of disease. The difference between the model with nonlocal diffusion and that with local diffusion is also discussed and nonlocal diffusion leads to more possibilities.
The novel coronavirus pneumonia (COVID-19) pandemic has caused enormous impacts around the world. Characterizing the risk dynamics for urgent epidemics such as COVID-19 is of great benefit to epidemic control and emergency management. This article presents a novel approach to characterizing the space-time risks of the COVID-19 epidemic. We analyzed the heavy-tailed distribution and spatial hierarchy of confirmed COVID-19 cases in 367 cities from 20 January to 12 April 2020, and population density data for 2019, and modelled two parameters, COVID-19 confirmed cases and population density, to measure the risk value of each city and assess the epidemic from the perspective of spatial and temporal changes. The evolution pattern of high-risk areas was assessed from a spatial and temporal perspective. The number of high-risk cities decreased from 57 in week 1 to 6 in week 12. The results show that the risk measurement model based on the head/tail breaks approach can describe the spatial and temporal evolution characteristics of the risk of COVID-19, and can better predict the risk trend of future epidemics in each city and identify the risk of future epidemics even during low incidence periods. Compared with the traditional risk assessment method model, it pays more attention to the differences in the spatial level of each city and provides a new perspective for the assessment of the risk level of epidemic transmission. It has generality and flexibility and provides a certain reference for the prevention of infectious diseases as well as a theoretical basis for government implementation strategies.
The COVID-19 epidemic has a profoundly negative impact on the lives of people all over the world. The measure of quarantine is one of the prominent means to suppress the spread of COVID-19. Therefore, we consider a novel susceptible-exposed-asymptomatic infected-symptomatic infected-quarantine-recovered (SEAIQR) model by adding quarantine subgroup to the SEAIR infectious disease model. Furthermore, to control the spread of the infectious disease, we design a fixed-time synergetic control (FTSC) approach based on the new model to develop three control strategies, which include quarantine of asymptomatic and symptomatic infected subgroups and vaccination of susceptible subgroup. In particular, the synergetic control method can provide non-chattering characteristics for the control system. In addition, the convergence speed of tracking reference signals can be improved with FTSC. Besides, the FTSC method can enable the latent subgroup, the asymptomatic infected subgroup and the symptomatic infection subgroup to converge to the desired value within a settling time without knowing the initial numbers of infection. Finally, numerical simulations are conducted to verify that the infectious disease can be controlled within a finite time with two groups of initial infection values.
In this study, we have proposed and analyzed a new COVID-19 and syphilis co-infection mathematical model with 10 distinct classes of the human population (COVID-19 protected, syphilis protected, susceptible, COVID-19 infected, COVID-19 isolated with treatment, syphilis asymptomatic infected, syphilis symptomatic infected, syphilis treated, COVID-19 and syphilis co-infected, and COVID-19 and syphilis treated) that describes COVID-19 and syphilis co-dynamics. We have calculated all the disease-free and endemic equilibrium points of single infection and co-infection models. The basic reproduction numbers of COVID-19, syphilis, and COVID-19 and syphilis co-infection models were determined. The results of the model analyses show that the COVID-19 and syphilis co-infection spread is under control whenever its basic reproduction number is less than unity. Moreover, whenever the co-infection basic reproduction number is greater than unity, COVID-19 and syphilis co-infection propagates throughout the community. The numerical simulations performed by MATLAB code using the ode45 solver justified the qualitative results of the proposed model. Moreover, both the qualitative and numerical analysis findings of the study have shown that protections and treatments have fundamental effects on COVID-19 and syphilis co-dynamic disease transmission prevention and control in the community.
This paper adopts a complex network approach for discussing the level of heterogeneity and cohesiveness among firms that have used a particular financial instrument — the so-called minibond. The nodes of the networks represent firms, and the weight of a link is assumed to be increasing with the similarity of the corresponding nodes/firms — where similarity is intended in terms of specific economic-financial characteristics of the firms. We assess the level of heterogeneity through the strength degree and the level of cohesiveness through the clustering coefficient. The empirical experiments are based on the paradigmatic case of the Italian reality, where minibonds have been and are currently efficiently used. The analysis reveals regularities and discrepancies among firms’ financial characteristics. Furthermore, the results suggest the potential identification of the main determinants of minibonds issuance.
The objective of this study is to propose a position-distance-related model based on complex networks for the H1N1 epidemic simulation. Situation updates of the H1N1 prevalenced in 2009 show that spreading of the epidemic virus is highly correlative with its position and distance as well. Then in the proposed simulation model, each node in the network is described with not only its edges but also its position and distance. Accordingly, two mechanisms called “growth with position” and “degree and distance based preferential attachment” are introduced in the proposed model that it establishes one connection with likelihood proportional to node’s degree and inversely proportional to the distance between two nodes. Beside the traditional node-growth mode, one called link-growth mode is also introduced. The main advantage of the proposed method is that it is one concise data-driven modeling based on complex networks. Simulation results utilizing the proposed link-growth mode and the traditional node-growth mode show that the two modes are equivalent to each other but from different perspectives. Moreover, compared to the traditional node-growth mode, the proposed link-growth mode is clear and concise.
This work provides a geometric version of the next-generation method for obtaining the basic reproduction number of an epidemiological model. More precisely, we generalize the concept of the basic reproduction number for the theory of Petri nets. The motivation comes from observing that any epidemiological model has the basic structures found in the SIR model of Kermack-McKendrick. These are three substructures, also given by three Petri nets inside, representing the susceptible population, the infection process, and the infected individuals. The five assumptions of the next-generation matrix given by van den Driessche-Watmough can be described geometrically using Petri nets. Thus, the next-generation matrix results in a matrix of flows between the infection compartments. Besides that, we construct a procedure that associates with a system of ordinary differential equations under certain assumptions, a Petri net that is minimal. In addition, we explain how Petri nets extend compartmental models to include vertical transmission.
While severe social-distancing measures have proven effective in slowing the coronavirus disease 2019 (COVID-19) pandemic, second-wave scenarios are likely to emerge as restrictions are lifted. Here we integrate anonymized, geolocalized mobility data with census and demographic data to build a detailed agent-based model of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) transmission in the Boston metropolitan area. We find that a period of strict social distancing followed by a robust level of testing, contact-tracing and household quarantine could keep the disease within the capacity of the healthcare system while enabling the reopening of economic activities. Our results show that a response system based on enhanced testing and contact tracing can have a major role in relaxing social-distancing interventions in the absence of herd immunity against SARS-CoV-2. An agent-based model of SARS-CoV-2 transmission shows that testing, contact tracing and household quarantine could keep new COVID-19 waves under control while allowing the reopening of the economy with minimal social-distancing interventions.
At the beginning of a COVID-19 infection, there is a period of time known as the exposed or latency period, before an infected person is capable of transmitting the infection to another person. We develop two differential equations models to account for this period. The first is a model that incorporates infected persons in the exposed class, before transmission is possible. The second is a model that incorporates a time delay in infected persons, before transmission is possible. We apply both models to the COVID-19 epidemic in China. We estimate the epidemiological parameters in the models, such as the transmission rate and the basic reproductive number, using data of reported cases. We thus evaluate the role of the exposed or latency period in the dynamics of a COVID-19 epidemic.
An SL1L2I1I2A1A2R epidemic model is formulated that describes the spread of an epidemic in a population. The model incorporates an Erlang distribution of times of sojourn in incubating, symptomatically and asymptomatically infectious compartments. Basic properties of the model are explored, with focus on properties important in the context of current COVID-19 pandemic.
Significance
The ongoing pandemic of COVID-19 challenges globalized societies. Scientific and technological cross-fertilization yields broad availability of georeferenced epidemiological data and of modeling tools that aid decisions on emergency management. To this end, spatially explicit models of the COVID-19 epidemic that include e.g. regional individual mobilities, the progression of social distancing, and local capacity of medical infrastructure provide significant information. Data-tailored spatial resolutions that model the disease spread geography can include details of interventions at the proper geographical scale. Based on them, it is possible to quantify the effect of local containment measures (like diachronic spatial maps of averted hospitalizations) and the assessment of the spatial and temporal planning of the needs of emergency measures and medical infrastructure as a major contingency planning aid.
In Italy, 128,948 confirmed cases and 15,887 deaths of people who tested positive for SARS-CoV-2 were registered as of 5 April 2020. Ending the global SARS-CoV-2 pandemic requires implementation of multiple population-wide strategies, including social distancing, testing and contact tracing. We propose a new model that predicts the course of the epidemic to help plan an effective control strategy. The model considers eight stages of infection: susceptible (S), infected (I), diagnosed (D), ailing (A), recognized (R), threatened (T), healed (H) and extinct (E), collectively termed SIDARTHE. Our SIDARTHE model discriminates between infected individuals depending on whether they have been diagnosed and on the severity of their symptoms. The distinction between diagnosed and non-diagnosed individuals is important because the former are typically isolated and hence less likely to spread the infection. This delineation also helps to explain misperceptions of the case fatality rate and of the epidemic spread. We compare simulation results with real data on the COVID-19 epidemic in Italy, and we model possible scenarios of implementation of countermeasures. Our results demonstrate that restrictive social-distancing measures will need to be combined with widespread testing and contact tracing to end the ongoing COVID-19 pandemic. A new epidemiological model, termed SIDARTHE, distinguishes between diagnosed and undiagnosed cases of SARS-CoV-2 infection, as well as modeling effects of social distancing and widespread testing, to predict possible outcomes of the COVID-19 epidemic in Italy.
We report temporal patterns of viral shedding in 94 patients with laboratory-confirmed COVID-19 and modeled COVID-19 infectiousness profiles from a separate sample of 77 infector–infectee transmission pairs. We observed the highest viral load in throat swabs at the time of symptom onset, and inferred that infectiousness peaked on or before symptom onset. We estimated that 44% (95% confidence interval, 25–69%) of secondary cases were infected during the index cases’ presymptomatic stage, in settings with substantial household clustering, active case finding and quarantine outside the home. Disease control measures should be adjusted to account for probable substantial presymptomatic transmission. Presymptomatic transmission of SARS-CoV-2 is estimated to account for a substantial proportion of COVID-19 cases.
We estimate the distribution of serial intervals for 468 confirmed cases of 2019 novel coronavirus disease reported in China as of February 8, 2020. The mean interval was 3.96 days (95% CI 3.53–4.39 days), SD 4.75 days (95% CI 4.46–5.07 days); 12.6% of case reports indicated presymptomatic transmission.
https://wwwnc.cdc.gov/eid/article/26/6/20-0357_article
Background
The coronavirus disease 2019 (COVID-19) is rapidly spreading in China and more than 30 countries over last two months. COVID-19 has multiple characteristics distinct from other infectious diseases, including high infectivity during incubation, time delay between real dynamics and daily observed number of confirmed cases, and the intervention effects of implemented quarantine and control measures.Methods
We develop a Susceptible, Un-quanrantined infected, Quarantined infected, Confirmed infected (SUQC) model to characterize the dynamics of COVID-19 and explicitly parameterize the intervention effects of control measures, which is more suitable for analysis than other existing epidemic models.ResultsThe SUQC model is applied to the daily released data of the confirmed infections to analyze the outbreak of COVID-19 in Wuhan, Hubei (excluding Wuhan), China (excluding Hubei) and four first-tier cities of China. We found that, before January 30, 2020, all these regions except Beijing had a reproductive number R > 1, and after January 30, all regions had a reproductive number R < 1, indicating that the quarantine and control measures are effective in preventing the spread of COVID-19. The confirmation rate of Wuhan estimated by our model is 0.0643, substantially lower than that of Hubei excluding Wuhan (0.1914), and that of China excluding Hubei (0.2189), but it jumps to 0.3229 after February 12 when clinical evidence was adopted in new diagnosis guidelines. The number of unquarantined infected cases in Wuhan on February 12, 2020 is estimated to be 3,509 and declines to 334 on February 21, 2020. After fitting the model with data as of February 21, 2020, we predict that the end time of COVID-19 in Wuhan and Hubei is around late March, around mid March for China excluding Hubei, and before early March 2020 for the four tier-one cities. A total of 80,511 individuals are estimated to be infected in China, among which 49,510 are from Wuhan, 17,679 from Hubei (excluding Wuhan), and the rest 13,322 from other regions of China (excluding Hubei). Note that the estimates are from a deterministic ODE model and should be interpreted with some uncertainty.Conclusions
We suggest that rigorous quarantine and control measures should be kept before early March in Beijing, Shanghai, Guangzhou and Shenzhen, and before late March in Hubei. The model can also be useful to predict the trend of epidemic and provide quantitative guide for other countries at high risk of outbreak, such as South Korea, Japan, Italy and Iran.
Spreading processes have been largely studied in the literature, both analytically and by means of large-scale numerical simulations. These processes mainly include the propagation of diseases, rumors and information on top of a given population. In the last two decades, with the advent of modern network science, we have witnessed significant advances in this field of research. Here we review the main theoretical and numerical methods developed for the study of spreading processes on complex networked systems. Specifically, we formally define epidemic processes on single and multilayer networks and discuss in details the main methods used to perform numerical simulations. Throughout the review, we classify spreading processes (disease and rumor models) into two classes according to the nature of time: (i) continuous-time and (ii) cellular automata approach, where the second one can be further divided into synchronous and asynchronous updating schemes. Our revision includes the heterogeneous mean-field, the quenched-mean field, and the pair quenched mean-field approaches, as well as their respective simulation techniques, emphasizing similarities and differences among the different techniques. The content presented here offers a whole suite of methods to study epidemic-like processes in complex networks, both for researchers without previous experience in the subject and for experts.
We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. At variance with what is observed in single-layer networks, we show that disease localization takes place on the layers and not on the nodes of a given layer. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: If the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we report on an interesting phenomenon, the barrier effect; i.e., for a three-layer configuration, when the layer with the lowest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems, opening new possibilities for the study of spreading processes.
Understanding, predicting, and controlling outbreaks of waterborne diseases are crucial goals of public health policies, but pose challenging problems because infection patterns are influenced by spatial structure and temporal asynchrony. Although explicit spatial modeling is made possible by widespread data mapping of hydrology, transportation infrastructure, population distribution, and sanitation, the precise condition under which a waterborne disease epidemic can start in a spatially explicit setting is still lacking. Here we show that the requirement that all the local reproduction numbers be larger than unity is neither necessary nor sufficient for outbreaks to occur when local settlements are connected by networks of primary and secondary infection mechanisms. To determine onset conditions, we derive general analytical expressions for a reproduction matrix , explicitly accounting for spatial distributions of human settlements and pathogen transmission via hydrological and human mobility networks. At disease onset, a generalized reproduction number {\hbox{ \Lambda }}_{\mathbf{0}} (the dominant eigenvalue of ) must be larger than unity. We also show that geographical outbreak patterns in complex environments are linked to the dominant eigenvector and to spectral properties of . Tests against data and computations for the 2010 Haiti and 2000 KwaZulu-Natal cholera outbreaks, as well as against computations for metapopulation networks, demonstrate that eigenvectors of provide a synthetic and effective tool for predicting the disease course in space and time. Networked connectivity models, describing the interplay between hydrology, epidemiology, and social behavior sustaining human mobility, thus prove to be key tools for emergency management of waterborne infections.
In recent years the increasing availability of computer power and informatics tools has enabled the gathering of reliable data quantifying the complexity of socio-technical systems. Data-driven computational models have emerged as appropriate tools to tackle the study of dynamical phenomena as diverse as epidemic outbreaks, information spreading and Internet packet routing. These models aim at providing a rationale for understanding the emerging tipping points and nonlinear properties that often underpin the most interesting characteristics of socio-technical systems. Here, using diffusion and contagion phenomena as prototypical examples, we review some of the recent progress in modelling dynamical processes that integrates the complex features and heterogeneities of real-world systems.
Using the susceptible-infected-susceptible model on unweighted and weighted networks, we consider the disease localization phenomenon. In contrast to the well-recognized point of view that diseases infect a finite fraction of vertices right above the epidemic threshold, we show that diseases can be localized on a finite number of vertices, where hubs and edges with large weights are centers of localization. Our results follow from the analysis of standard models of networks and empirical data for real-world networks.
Many epidemic processes in networks spread by stochastic contacts among their connected vertices. There are two limiting cases widely analyzed in the physics literature, the so-called contact process (CP) where the contagion is expanded at a certain rate from an infected vertex to one neighbor at a time, and the reactive process (RP) in which an infected individual effectively contacts all its neighbors to expand the epidemics. However, a more realistic scenario is obtained from the interpolation between these two cases, considering a certain number of stochastic contacts per unit time. Here we propose a discrete-time formulation of the problem of contact-based epidemic spreading. We resolve a family of models, parameterized by the number of stochastic contact trials per unit time, that range from the CP to the RP. In contrast to the common heterogeneous mean-field approach, we focus on the probability of infection of individual nodes. Using this formulation, we can construct the whole phase diagram of the different infection models and determine their critical properties. Comment: 6 pages, 4 figures. Europhys Lett (in press 2010)
The Internet has a very complex connectivity recently modeled by the class of scale-free networks. This feature, which appears to be very efficient for a communications network, favors at the same time the spreading of computer viruses. We analyze real data from computer virus infections and find the average lifetime and persistence of viral strains on the Internet. We define a dynamical model for the spreading of infections on scale-free networks, finding the absence of an epidemic threshold and its associated critical behavior. This new epidemiological framework rationalizes data of computer viruses and could help in the understanding of other spreading phenomena on communication and social networks.
Random scale-free networks have the peculiar property of being prone to the spreading of infections. Here we provide for the susceptible-infected-susceptible model an exact result showing that a scale-free degree distribution with diverging second moment is a sufficient condition to have null epidemic threshold in unstructured networks with either assortative or disassortative mixing. Degree correlations result therefore irrelevant for the epidemic spreading picture in these scale-free networks. The present result is related to the divergence of the average nearest neighbor's degree, enforced by the degree detailed balance condition.
In this paper, a new Susceptible-Exposed-Symptomatic Infectious-Asymptomatic Infectious-Quarantined-Hospitalized-Recovered-Dead (SEIDIUQHRD) deterministic compartmental model has been proposed and calibrated for interpreting the transmission dynamics of the novel coronavirus disease (COVID-19). The purpose of this study is to give a tentative prediction of the epidemic peak for Russia, Brazil, India and Bangladesh which could become the next COVID-19 hotspots in no time by using a Trust-region-reflective (TRR) algorithm which one of the well-known real data fitting techniques. Based on the publicly available epidemiological data from late January until 10 May, it has been estimated that the number of daily new symptomatic infectious cases for the above mentioned countries could reach the peak around the beginning of June with the peak size of ∼ 15, 774 (95% CI, 12,814-16,734) symptomatic infectious cases in Russia, ∼ 26, 449 (95% CI, 25,489-31,409) cases in Brazil, ∼ 9, 504 (95% CI, 8,378-13,630) cases in India and ∼ 2, 209 (95% CI, 2,078-2,840) cases in Bangladesh. As of May 11, 2020, incorporating the infectiousness capability of asymptomatic carriers, our analysis estimates the value of the basic reproduction number (R0) as of May 11, 2020 was found to be ∼ 4.234 (95% CI, 3.764-4.7) in Russia, ∼ 5.347 (95% CI, 4.737-5.95) in Brazil, ∼ 5.218 (95% CI, 4.56-5.81)in India, ∼ 4.649 (95% CI, 4.17-5.12) in the United Kingdom and ∼ 3.53 (95% CI, 3.12-3.94) in Bangladesh. Moreover, Latin hypercube sampling-partial rank correlation coefficient (LHS-PRCC) which is a global sensitivity analysis (GSA) method is applied to quantify the uncertainty of our model mechanisms, which elucidates that for Russia, the recovery rate of undetected asymptomatic carriers, the rate of getting home-quarantined or self-quarantined and the transition rate from quarantined class to susceptible class are the most influential parameters, whereas the rate of getting home-quarantined or self-quarantined and the inverse of the COVID-19 incubation period are highly sensitive parameters in Brazil, India, Bangladesh and the United Kingdom which could significantly affect the transmission dynamics of the novel coronavirus. Our analysis also suggests that relaxing social distancing restrictions too quickly could exacerbate the epidemic outbreak in the above-mentioned countries.
We propose a compartmental mathematical model for the spread of the COVID-19 disease with special focus on the transmissibility of super-spreaders individuals. We compute the basic reproduction number threshold, we study the local stability of the disease free equilibrium in terms of the basic reproduction number, and we investigate the sensitivity of the model with respect to the variation of each one of its parameters. Numerical simulations show the suitability of the proposed COVID-19 model for the outbreak that occurred in Wuhan, China.
Background:
A novel human coronavirus, severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), was identified in China in December 2019. There is limited support for many of its key epidemiologic features, including the incubation period for clinical disease (coronavirus disease 2019 [COVID-19]), which has important implications for surveillance and control activities.
Objective:
To estimate the length of the incubation period of COVID-19 and describe its public health implications.
Design:
Pooled analysis of confirmed COVID-19 cases reported between 4 January 2020 and 24 February 2020.
Setting:
News reports and press releases from 50 provinces, regions, and countries outside Wuhan, Hubei province, China.
Participants:
Persons with confirmed SARS-CoV-2 infection outside Hubei province, China.
Measurements:
Patient demographic characteristics and dates and times of possible exposure, symptom onset, fever onset, and hospitalization.
Results:
There were 181 confirmed cases with identifiable exposure and symptom onset windows to estimate the incubation period of COVID-19. The median incubation period was estimated to be 5.1 days (95% CI, 4.5 to 5.8 days), and 97.5% of those who develop symptoms will do so within 11.5 days (CI, 8.2 to 15.6 days) of infection. These estimates imply that, under conservative assumptions, 101 out of every 10 000 cases (99th percentile, 482) will develop symptoms after 14 days of active monitoring or quarantine.
Limitation:
Publicly reported cases may overrepresent severe cases, the incubation period for which may differ from that of mild cases.
Conclusion:
This work provides additional evidence for a median incubation period for COVID-19 of approximately 5 days, similar to SARS. Our results support current proposals for the length of quarantine or active monitoring of persons potentially exposed to SARS-CoV-2, although longer monitoring periods might be justified in extreme cases.
Primary funding source:
U.S. Centers for Disease Control and Prevention, National Institute of Allergy and Infectious Diseases, National Institute of General Medical Sciences, and Alexander von Humboldt Foundation.
We propose a novel epidemic model based on two-layered multiplex networks to explore the influence of positive and negative preventive information on epidemic propagation. In the model, one layer represents a social network with positive and negative preventive information spreading competitively, while the other one denotes the physical contact network with epidemic propagation. The individuals who are aware of positive prevention will take more effective measures to avoid being infected than those who are aware of negative prevention. Taking the microscopic Markov chain (MMC) approach, we analytically derive the expression of the epidemic threshold for the proposed epidemic model, which indicates that the diffusion of positive and negative prevention information, as well as the topology of the physical contact network have a significant impact on the epidemic threshold. By comparing the results obtained with MMC and those with the Monte Carlo (MC) simulations, it is found that they are in good agreement, but MMC can well describe the dynamics of the proposed model. Meanwhile, through extensive simulations, we demonstrate the impact of positive and negative preventive information on the epidemic threshold, as well as the prevalence of infectious diseases. We also find that the epidemic prevalence and the epidemic outbreaks can be suppressed by the diffusion of positive preventive information and be promoted by the diffusion of negative preventive information.
The book is a comprehensive, self-contained introduction to the mathematical modeling and analysis of disease transmission models. It includes (i) an introduction to the main concepts of compartmental models including models with heterogeneous mixing of individuals and models for vector-transmitted diseases, (ii) a detailed analysis of models for important specific diseases, including tuberculosis, HIV/AIDS, influenza, Ebola virus disease, malaria, dengue fever and the Zika virus, (iii) an introduction to more advanced mathematical topics, including age structure, spatial structure, and mobility, and (iv) some challenges and opportunities for the future.
There are exercises of varying degrees of difficulty, and projects leading to new research directions. For the benefit of public health professionals whose contact with mathematics may not be recent, there is an appendix covering the necessary mathematical background. There are indications which sections require a strong mathematical background so that the book can be useful for both mathematical modelers and public health professionals.
A general formalism is introduced to allow the steady state of non-Markovian processes on networks to be reduced to equivalent Markovian processes on the same substrates. The example of an epidemic spreading process is considered in detail, where all the non-Markovian aspects are shown to be captured within a single parameter, the effective infection rate. Remarkably, this result is independent of the topology of the underlying network, as demonstrated by numerical simulations on two-dimensional lattices and various types of random networks. Furthermore, an analytic approximation for the effective infection rate is introduced, which enables the calculation of the critical point and of the critical exponents for the non-Markovian dynamics.
Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mech-anisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
In recent years the research community has accumulated overwhelming evidence
for the emergence of complex and heterogeneous connectivity patterns in a wide
range of biological and socio-technical systems. The complex properties of real
world networks have a profound impact on the behavior of equilibrium and
non-equilibrium phenomena occurring in various systems, and the study of
epidemic spreading is central to our understanding of the unfolding of
dynamical processes in complex networks. The theoretical analysis of epidemic
spreading in heterogeneous networks requires the development of novel
analytical frameworks, and it has produced results of conceptual and practical
relevance. Here we present a coherent and comprehensive review of the vast
research activity concerning epidemic processes, detailing the successful
theoretical approaches as well as making their limits and assumptions clear.
Physicists, epidemiologists, computer and social scientists share a common
interest in studying epidemic spreading and rely on very similar models for the
description of the diffusion of pathogens, knowledge, and innovation. For this
reason, while we focus on the main results and the paradigmatic models in
infectious disease modeling, we also present the major results concerning
generalized social contagion processes. Finally we outline the research
activity at the forefront in the study of epidemic spreading in co-evolving and
time-varying networks.
Exploiting the power of the expectation operator and indicator (or Bernoulli)
random variables, we present the exact governing equations for both the SIR and
SIS epidemic models on \emph{networks}. Although SIR and SIS are basic epidemic
models, deductions from their exact stochastic equations \textbf{without}
making approximations (such as the common mean-field approximation) are scarce.
An exact analytic solution of the governing equations is highly unlikely to be
found (for any network) due to the appearing pair (and higher order)
correlations. Nevertheless, the maximum average fraction of infected
nodes in both SIS and SIR can be written as a quadratic form of the graph's
Laplacian. Only for regular graphs, the expression for the maximum of
can be simplied to exhibit the explicit dependence on the spectral radius. From
our new Laplacian expression, we deduce a general \textbf{upper} bound for the
epidemic SIS threshold in any graph.
Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number R0, the contact number σ, and the replacement number R are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for R0 are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of R0 and σ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.
Based on the SIR (Susceptible-Infected-Removed) model, we propose a novel epidemic model to investigate the impact of infection delay and propagation vector on the spreading behaviors in complex networks. Mean-field approximations and extensive numerical simulations indicate that the infection delay and propagation vector can largely reduce the critical threshold and promote the outbreak of epidemics, and even lead to the case that the infectious diseases transform from the disease-free state to endemic one. The current results are greatly instructive for us to further understand the epidemic spreading and design some effective prevention and containment strategies to fight the epidemics.
We investigate the effects of delaying the time to recovery (delayed recovery) and of nonuniform transmission on the propagation of diseases on structured populations. Through a mean-field approximation and large-scale numerical simulations, we find that postponing the transition from the infectious to the recovered states can largely reduce the epidemic threshold, therefore promoting the outbreak of epidemics. On the other hand, if we consider nonuniform transmission among individuals, the epidemic threshold increases, thus inhibiting the spreading process. When both mechanisms are at work, the latter might prevail, hence resulting in an increase of the epidemic threshold with respect to the standard case, in which both ingredients are absent. Our findings are of interest for a better understanding of how diseases propagate on structured populations and to a further design of efficient immunization strategies.
Most studies on susceptible-infected-susceptible epidemics in networks implicitly assume Markovian behavior: the time to infect a direct neighbor is exponentially distributed. Much effort so far has been devoted to characterize and precisely compute the epidemic threshold in susceptible-infected-susceptible Markovian epidemics on networks. Here, we report the rather dramatic effect of a nonexponential infection time (while still assuming an exponential curing time) on the epidemic threshold by considering Weibullean infection times with the same mean, but different power exponent α. For three basic classes of graphs, the Erdős-Rényi random graph, scale-free graphs and lattices, the average steady-state fraction of infected nodes is simulated from which the epidemic threshold is deduced. For all graph classes, the epidemic threshold significantly increases with the power exponents α. Hence, real epidemics that violate the exponential or Markovian assumption can behave seriously differently than anticipated based on Markov theory.
The influence of the network characteristics on the virus spread is analyzed in a new-the N -intertwined Markov chain-model, whose only approximation lies in the application of mean field theory. The mean field approximation is quantified in detail. The N -intertwined model has been compared with the exact 2N-state Markov model and with previously proposed ldquohomogeneousrdquo or ldquolocalrdquo models. The sharp epidemic threshold tauc , which is a consequence of mean field theory, is rigorously shown to be equal to tauc = 1/(lambdamax( A )) , where lambdamax( A ) is the largest eigenvalue-the spectral radius-of the adjacency matrix A . A continued fraction expansion of the steady-state infection probability at node j is presented as well as several upper bounds.
Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.
Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
We study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks. In networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null. On the contrary, on a wide range of scale-free networks we observe the absence of an epidemic threshold and its associated critical behavior. This implies that scale-free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess. These results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks.
The study of social networks, and in particular the spread of disease on networks, has attracted considerable recent attention in the physics community. In this paper, we show that a large class of standard epidemiological models, the so-called susceptible/infective/removed (SIR) models can be solved exactly on a wide variety of networks. In addition to the standard but unrealistic case of fixed infectiveness time and fixed and uncorrelated probability of transmission between all pairs of individuals, we solve cases in which times and probabilities are nonuniform and correlated. We also consider one simple case of an epidemic in a structured population, that of a sexually transmitted disease in a population divided into men and women. We confirm the correctness of our exact solutions with numerical simulations of SIR epidemics on networks.
How will a virus propagate in a real network? Does an epidemic threshold exist for a finite graph? How long does it take to disinfect a network given particular values of infection rate and virus death rate? We answer the first question by providing equations that accurately model virus propagation in any network including real and synthesized network graphs. We propose a general epidemic threshold condition that applies to arbitrary graphs: we prove that, under reasonable approximations, the epidemic threshold for a network is closely related to the largest eigenvalue of its adjacency matrix. Finally, for the last question, we show that infections tend to zero exponentially below the epidemic threshold. We show that our epidemic threshold model subsumes many known thresholds for special-case graphs (e.g., Erdos-Renyi, BA power-law, homogeneous); we show that the threshold tends to zero for infinite power-law graphs. We show that our threshold condition holds for arbitrary graphs.
Got the flu (or mumps
- B A Prakash
- D Chakrabarti
- M Faloutsos
- N Valler
- C Faloutsos
Prakash B.A., Chakrabarti D., Faloutsos M., Valler N., Faloutsos C.. Got the flu
(or mumps)? Check the eigenvalue! arXiv:10 040 060 2010;.
Expected impact of reopening schools after lockdown on COVID-19 epidemic in Île-de-France
- Di Domenico
- L Pullano
- G Sabbatini
- C E Boëlle
- P-Y Colizza
Di Domenico L, Pullano G, Sabbatini CE, Boëlle P-Y, Colizza V. Expected impact
of reopening schools after lockdown on COVID-19 epidemic in Île-de-France.
medRxiv 2020.
Modeling the impact of social distancing, testing, contact tracing and household quarantine on second-wave scenarios of the COVID-19 epidemic
- Aleta A Martin-Corral
- D Piontti
- A P Ajelli
- M Litvinova
- M Chinazzi
Aleta A, Martin-Corral D, y Piontti AP, Ajelli M, Litvinova M, Chinazzi M,
et al. Modeling the impact of social distancing, testing, contact tracing and
household quarantine on second-wave scenarios of the COVID-19 epidemic.
medRxiv 2020.
Epidemic analysis of COVID-19 in China by dynamical modeling
- L Peng
- W Yang
- D Zhang
- C Zhuge
- L Hong
Peng L., Yang W., Zhang D., Zhuge C., Hong L.. Epidemic analysis of COVID-19
in China by dynamical modeling. arXiv:200206563 2020;.
Modeling the impact of social distancing, testing, contact tracing and household quarantine on second-wave scenarios of the COVID-19 epidemic
- Aleta