SEIR Epidemic Model Research Articles

Stability analysis of the SEIRS epidemic model with infectious dynamics during latent and infectious periods

Coronary pneumonia caused by the SARS-CoV-2 virus was one of the most significant public health threats in recent years. In this paper, we develop and investigate an SEIRS epidemic model that incorporates infectivity during both the latent and infectious stages to characterize the transmission dynamics of COVID-19. By calculating the basic reproduction number (R0) and applying monotonic dynamical system theory along with geometric methods, we validate the threshold theorem. Our analysis demonstrates that the disease-free equilibrium is globally stable when R0 < 1, while the endemic equilibrium becomes globally stable when R0 > 1.

Analysis and Optimal Control of a Two-Strain SEIR Epidemic Model with Saturated Treatment Rate

In this paper, we conducted a study on the optimal control problem of an epidemic model which consists of two strain with different types of incidence rates: bilinear and non-monotonic. We also considered use of the saturation treatment function. Two basic regeneration numbers are calculated from the epidemic model, which are denoted as R1 and R2. The global stability of the disease-free equilibrium point was studied by the Lyapunov method, and it was proved that the disease-free equilibrium point is globally asymptotically stable when R1 and R2 are less than one. Finally, we formulated a time-dependent optimal control problem by Pontryagin’s maximum principle. Numerical simulations were performed to establish the effects of model parameters for disease transmission as well as the effects of control.

Untangling the mobility dynamics in a patchy environment: An SEIR epidemic model with vertical transmission

Congenital Rubella, AIDS, and other human diseases may pass from mother to fetus through the placenta. On the other side, insects and plants often transmit vertically by their eggs/seeds. Here, we study an SEIR epidemic model with vertical transmission in a multi‐patch context to explore the spread of disease in a population divided into distinct regions. The local and global stability of disease‐free equilibrium for the n‐patch model system has been determined in terms of the basic reproduction number . Moreover, we discuss the existence and stability of endemic equilibrium for different cases in two symmetric regions. Our findings indicate that when is less than one, the disease‐free equilibrium is globally asymptotically stable, whereas when is greater than one, it is unstable. As a result, for , the model system has an unique endemic equilibrium that is always locally asymptotically stable. The dynamics of the proposed model system depending on the distribution of infected individuals across various regions have been studied. The findings suggest that the movements from high disease risk regions to low disease risks may increase the number of the infected population in the community. The value of could be enhanced by increasing the values of any vertical transmission parameter. However, the changes in the values of are relatively minor. Our findings also imply that while migrations of all individuals across symmetric regions do not leave any impact on the equilibrium state of the infected population, they do affect the trajectory of the solution to reach a steady‐state. The flow of people between the two symmetrical regions could not alter the endemicity.

A spatio-temporal infection epidemic model with fractional order, general incidence, and vaccination analysis

The paper investigates a spatio-temporal SEIR epidemic model on a global level with fractional order. It employs four partial differential equations incorporating fractional derivatives and account for diffusion to characterize infection dynamics. By applying fixed-point theorem results, the paper establishes the existence, uniqueness, and boundedness of the solution. Equilibrium points are determined based on vaccination values and the basic reproduction number R0. Global stability of each equilibrium is confirmed using the Lyapunov direct method. Through simulations with a predictor–corrector algorithm, key insights into epidemiological dynamics are provided, elucidating the impact of vaccination on reducing disease transmission and altering R0. Trajectories of various compartments closely align with theoretical equilibrium points, affirming the model’s predictive precision. Furthermore, simulations indicate the potential for attaining disease-free equilibria with heightened vaccination rates, underscoring the pivotal role of vaccination strategies in epidemic control and disease eradication. It was shown that the fractional derivative order has no effect on the equilibrium stability but rather only on the convergence speed towards the equilibrium points.

Global stability analysis, stabilization and synchronization of an SEIR epidemic model

In this paper, we investigate an SEIR epidemic model in which recruitment is assumed to be regulated by a constant function. The SEIR model is a widely-used epidemiological framework that categorises the population into four distinct groups: Susceptible, Exposed, Infectious, and Removed. It employs a system of differential equations to describe the transitions of individuals among these groups, making it a valuable tool for understanding and modelling the dynamics of infectious diseases. Mathematical analysis is used to study the dynamic behaviour of this model. We first determine the basic reproduction number, denoted as R 0 , which governs the dynamics of the SEIR model. Our findings reveal that when R 0 is less than 1, the unique disease-free equilibrium is asymptotically stable, and there is no endemic equilibrium. However, when R 0 exceeds 1, the disease-free equilibrium becomes unstable, and a unique, asymptotically stable endemic equilibrium emerges. Furthermore, by constructing appropriate Lyapunov functions, we have studied the global asymptotic stability of the two equilibrium points. On the other hand, we have used the backstepping method to design a control law that stabilises the SEIR model towards the disease-free equilibrium, which means the eradication of the disease from the target population. Next, we have used the same method to synchronise the SEIR drive system and the SEIR response system. This method is used to stabilise the synchronisation error dynamics arising from the mismatch between the drive and response systems. Finally, we presented numerical simulations to illustrate the theoretical results.

Time periodic traveling wave solutions of a time-periodic reaction–diffusion SEIR epidemic model with periodic recruitment

This paper focuses on the existence and nonexistence of a time periodic traveling wave solution of a time-periodic reaction–diffusion SEIR epidemic model. The main feature of the model is the possible deficiency of the classical comparison principle such that many known results do not directly work. If the basic reproduction number of the model, denoted by R0, is larger than one, there exists a minimal wave speed c∗>0 satisfying for each c>c∗, the system admits a nontrivial time periodic traveling wave solution with wave speed c and for c<c∗, there exists no nontrivial time periodic traveling waves such that the system; if R0<1, the system admits no nontrivial time periodic traveling waves.

A simple approach for studying stability properties of an SEIRS epidemic model

Abstract In this work, we study stability properties of a well-known integer-order SEIRS model with nonlinear incidence and vertical transmission. Firstly, we introduce a simple approach to the analysis of global asymptotic stability (GAS) of the integer-order model. This approach is based on general quadratic Lyapunov functions and characteristic of quadratic forms associated with real matrices. The result is that the GAS of disease-free and disease-endemic equilibrium points is completely established. This provides an important improvement for results constructed in two previous works. Secondly, we generalize the integer-order SEIRS model by considering it in the context of the Caputo fractional-order derivative. After that, the present approach is utilized to investigate the GAS of the proposed fractional-order model. As an important consequence, not only the GAS but also the uniform stability of the fractional-order model are determined fully. Therefore, the applicability of the approach is shown. Finally, a series of numerical experiments is conducted to illustrate and support the theoretical findings.

An SEIR network epidemic model with manual and digital contact tracing allowing delays

We consider an SEIR epidemic model on a network also allowing random contacts, where recovered individuals could either recover naturally or be diagnosed. Upon diagnosis, manual contact tracing is triggered such that each infected network contact is reported, tested and isolated with some probability and after a random delay. Additionally, digital tracing (based on a tracing app) is triggered if the diagnosed individual is an app-user, and then all of its app-using infectees are immediately notified and isolated. The early phase of the epidemic with manual and/or digital tracing is approximated by different multi-type branching processes, and three respective reproduction numbers are derived. The effectiveness of both contact tracing mechanisms is numerically quantified through the reduction of the reproduction number. This shows that app-using fraction plays an essential role in the overall effectiveness of contact tracing. The relative effectiveness of manual tracing compared to digital tracing increases if: more of the transmission occurs on the network, when the tracing delay is shortened, and when the network degree distribution is heavy-tailed. For realistic values, the combined tracing case can reduce R0 by 20%–30%, so other preventive measures are needed to reduce the reproduction number down to 1.2–1.4 for contact tracing to make it successful in avoiding big outbreaks.

Dynamics of simplicial SEIRS epidemic model: global asymptotic stability and neural Lyapunov functions.

The transmission of infectious diseases on a particular network is ubiquitous in the physical world. Here, we investigate the transmission mechanism of infectious diseases with an incubation period using a networked compartment model that contains simplicial interactions, a typical high-order structure. We establish a simplicial SEIRS model and find that the proportion of infected individuals in equilibrium increases due to the many-body connections, regardless of the type of connections used. We analyze the dynamics of the established model, including existence and local asymptotic stability, and highlight differences from existing models. Significantly, we demonstrate global asymptotic stability using the neural Lyapunov function, a machine learning technique, with both numerical simulations and rigorous analytical arguments. We believe that our model owns the potential to provide valuable insights into transmission mechanisms of infectious diseases on high-order network structures, and that our approach and theory of using neural Lyapunov functions to validate model asymptotic stability can significantly advance investigations on complex dynamics of infectious disease.

Modeling infectious respiratory diseases considering fear effect and latent period

This paper introduces a fractional-order (FO) SEIR epidemic model for respiratory diseases considering a non-human class P(t) for pathogens to study the effects of fear and awareness programs on disease dynamics. Further, using the basic reproduction number ℛ0α, the criteria for the extinction or persistence of the disease is established. Also, the conditions for Hopf bifurcation are derived, considering both FO (α) and rate of pathogens η as the bifurcation parameters. In addition, a detailed numerical simulation is performed to substantiate our theoretical results. The study of transmission dynamics of Tuberculosis (TB), a particular example of respiratory disease, is carried out in reference to the United States (US). Finally, we have estimated model parameters with the help of actual TB data from the US and then predicted the TB dynamics and disease control. It is pointed out that the fractional order can reduce the complexity of the model and better predict the dynamics of TB in the US than the integer order.

SEIR Epidemic Model of the Spread of Tuberculosis in Samarinda City with the Addition of Vaccination Parameters

Tuberculosis (TB) is a contagious, dangerous disease that infects the human body through the respiratory tract. The city of Samarinda itself is the city with the second highest rate of tuberculosis transmission in East Kalimantan. The aim of this research is to build a SEIR mathematical model of the spread of tuberculosis, analyze the stability of the model, and simulate the model. This research uses the Runge-Kutta method, which has high accuracy in estimating solutions and is relatively stable. This data is secondary data obtained through BPS as well as articles from previous researchers. The results of the research showed that the percentage of increase in the spread of the virus in the susceptible population (S) during the first week was 2,048%, then the 4th week to the 100th day decreased by approximately 164,100%, then the percentage of the population that was exposed but did not spread it (E) decreased in the first week, namely 34,525%, then in the 4th week to the 100th day there was a very significant decrease, namely 7,600%, then the percentage of the population infected and infectious (I) in the first week rose to 19,138%, then in the 4th week to the 2nd week -100 experienced a decrease of approximately 716,900%, and finally, the percentage of the population who recovered in the first week began to increase due to the very influential vaccine, namely increasing by 26,860%, then the 4th week to the 100th week also experienced a significant increase, namely by 81.681%.

Final size of an n-group SEIR epidemic model with nonlinear incidence rate

Calculation of the final infection size has become a topic of significant interest in recent years. Despite considerable progress, determining the final infection size in a heterogeneous infectious disease model with nonlinear incidence rate on short-time scales remains a challenging problem. In this paper, we investigate a heterogeneous [Formula: see text] epidemic model with nonlinear incidence rate. We establish both the existence and uniqueness of the solution regarding final size, and based on which, we are able to introduce a computational algorithm to calculate the final infection size. Furthermore, we apply our findings to study the early phase of the COVID-19 endemic in New York County and present a numerical simulation to illustrate the practical implications of our approach.

Origins of the problematic E in SEIR epidemic models

During the COVID-19 pandemic, over one thousand papers were published on “Susceptible-Exposed-Infectious-Removed” (SEIR) epidemic computational models. The English word “exposed” in its vernacular and public health usage means a state of having been in contact with an infectious individual, but not necessarily infected. In contrast, the term “exposed” in SEIR modeling usage typically stands for a state of already being infected but not yet being infectious to others, a state more properly termed “latently infected.” In public health language, “exposed” means possibly infected, yet in SEIR modeling language, “exposed” means already infected. This paper retraces the conceptual and mathematical origins of this terminological disconnect and concludes that epidemic modelers should consider using the “SLIR” notational short-hand (L for Latent) instead of SEIR.

Numerical approximation of fractional SEIR epidemic model of measles and smoking model by using Fibonacci wavelets operational matrix approach

In the present article, we have considered two essential models (The epidemic model of measles and the smoking model). Across the globe, the primary cause of health problems is smoking. Measles can be controlled in infectious populations using the mathematical model representing the direct transmission of infectious diseases. The Caputo fractional derivative operator of order α∈[0,1] is used to determine the solution of the system of fractional differential equations in both models. We employed the Fibonacci wavelets operational matrix of integration and developed the Fibonacci wavelet collocation method (FWCM) to solve a nonlinear coupled system of fractional differential equations (NCSFDEs). The proposed approach transforms the given model into a system of algebraic equations. The Newton-Raphson method is considered to extract the unknown coefficient values. Numerical outcomes are obtained to illustrate the simplicity and effectiveness of the proposed method. Graphs and tables show how consistently and effectively the developed strategy works. Mathematical software called Mathematica has been used to perform all calculations. Theorems explain the convergence of this approach.

On cognitive epidemic models: spatial segregation versus nonpharmaceutical interventions.

Fick's law and the Fokker-Planck law of diffusion are applied to manifest the cognitive dispersal of individuals in two reaction-diffusion SEIR epidemic models, where the disease transmission is illustrated by nonlocal infection mechanisms in heterogeneous environments. Building upon the well-posedness of solutions, threshold dynamics are discussed in terms of the basic reproduction numbers for the two cognitive epidemic models. The numerical investigation reveals that the Fokker-Planck law can better describe the diffusion of individuals by taking different dispersal strategies of exposed individuals in our cognitive epidemic models, and provides some insights on spatial segregation and nonpharmaceutical interventions: (i) spatial segregation occurs in the random diffusion model when the nonlocal infection radius is small, while it appears in the symmetric diffusion model when the radius is large; (ii) nonpharmaceutical interventions on restricting the dispersal of exposed and infected individuals do not contribute to reducing the infection proportion, but rather eliminate the disease in a region, which expands as the nonlocal infection radius increases. We additionally find that the final infection size in the random diffusion model is significantly smaller than that in the symmetric diffusion model and decreases as the nonlocal infection radius increases.

Dynamics of a Stochastic SEIR Epidemic Model with Vertical Transmission and Standard Incidence

A stochastic SEIR epidemic model with standard incidence and vertical transmission was developed in this work. The primary goal of this study was to determine whether stochastic environmental disturbances affect dynamic features of the epidemic model. The existence, uniqueness, and boundedness of global positive solutions are stated. A threshold was determined for the extinction of the infectious disease. After that, the existence and uniqueness of an ergodic stationary distribution were verified by determining the correct Lyapunov function. Ultimately, theoretical outcomes of numerical simulations are shown.

Relaxation Oscillation in SEIR Epidemic Models with the Intrinsic Growth Rate

The periodic oscillation transmission of infectious diseases is widespread, deep understanding of this periodic pattern and exploring the generation mechanism, and identifying the specific factors that lead to such periodic outbreaks, which are of very importanceto predict and control the spread of infectious diseases. In this study, to further reveal the mathematical mechanism of spontaneous generation of periodic oscillation solution, we investigate a type of SEIR epidemic model with a small intrinsic growth rate. By utilizing the singular perturbation theory and center manifold theorem, we extend the relaxation oscillation of three‐dimensional SIR models to the four‐dimensional SEIR models and prove the existence of stable relaxation oscillation with a large amplitude in the model. Numerical simulations are performed to verify our theoretical results. The results presented in this study provide a new idea for the study of the intrinsic mechanism of periodic oscillation in epidemiology, enrich the dynamics of epidemic models, and deepen the understanding of the global dynamics of these models.

An SEIR epidemic model's global analysis that incorporates the bi-linear incidence rate with treatment function

An SEIR epidemic model's global analysis that incorporates the bi-linear incidence rate with treatment function

Quantitative analysis of a fractional order of the $ SEI_{c}\, I_{\eta} VR $ epidemic model with vaccination strategy

&lt;abstract&gt;&lt;p&gt;This work focused on studying the effect of vaccination rate $ \kappa $ on reducing the outbreak of infectious diseases, especially if the infected individuals do not have any symptoms. We employed the fractional order derivative in this study since it has a high degree of accuracy. Recently, a lot of scientists have been interested in fractional-order models. It is considered a modern direction in the mathematical modeling of epidemiology systems. Therefore, a fractional order of the SEIR epidemic model with two types of infected groups and vaccination strategy was formulated and investigated in this paper. The proposed model includes the following classes: susceptible $ \mathrm{S}(t) $, exposed $ \mathrm{E}(t) $, asymptomatic infected $ \mathrm{I_{c}}(t) $, symptomatic infected $ \mathrm{I_{\eta}}(t) $, vaccinated $ \mathrm{V}(t) $, and recovered $ \mathrm{R}(t) $. We began our study by creating the existence, non-negativity, and boundedness of the solutions of the proposed model. Moreover, we established the basic reproduction number $ \mathcal{R}_{0} $, that was used to examine the existence and stability of the equilibrium points for the presented model. By creating appropriate Lyapunov functions, we proved the global stability of the free-disease equilibrium point and endemic equilibrium point. We concluded that the free-disease equilibrium point is globally asymptotically stable (GAS) when $ \mathcal{R}_{0}\, \leq \, 1 $, while the endemic equilibrium point is GAS if $ \mathcal{R}_{0} &amp;gt; 1 $. Therefore, we indicated the increasing vaccination rate $ \kappa $ leads to reducing $ \mathcal{R}_0 $. These findings confirm the important role of vaccination rate $ \kappa $ in fighting the spread of infectious diseases. Moreover, the numerical simulations were introduced to validate theoretical results that are given in this work by applying the predictor-corrector PECE method of Adams-Bashforth-Moulton. Further more, the impact of the vaccination rate $ \kappa $ was explored numerically and we found that, as $ \kappa $ increases, the $ \mathcal{R}_{0} $ is decreased. This means the vaccine can be useful in reducing the spread of infectious diseases.&lt;/p&gt;&lt;/abstract&gt;

Analysis of the fractional SEIR epidemic model with Caputo derivative via resolvents operators and numerical scheme

Analysis of the fractional SEIR epidemic model with Caputo derivative via resolvents operators and numerical scheme