Susceptible-exposed-infected-recovered-susceptible Research Articles

Control, bi-stability, and preference for chaos in time-dependent vaccination campaign.

In this work, effects of constant and time-dependent vaccination rates on the Susceptible-Exposed-Infected-Recovered-Susceptible (SEIRS) seasonal model are studied. Computing the Lyapunov exponent, we show that typical complex structures, such as shrimps, emerge for given combinations of a constant vaccination rate and another model parameter. In some specific cases, the constant vaccination does not act as a chaotic suppressor and chaotic bands can exist for high levels of vaccination (e.g., >0.95). Moreover, we obtain linear and non-linear relationships between one control parameter and constant vaccination to establish a disease-free solution. We also verify that the total infected number does not change whether the dynamics is chaotic or periodic. The introduction of a time-dependent vaccine is made by the inclusion of a periodic function with a defined amplitude and frequency. For this case, we investigate the effects of different amplitudes and frequencies on chaotic attractors, yielding low, medium, and high seasonality degrees of contacts. Depending on the parameters of the time-dependent vaccination function, chaotic structures can be controlled and become periodic structures. For a given set of parameters, these structures are accessed mostly via crisis and, in some cases, via period-doubling. After that, we investigate how the time-dependent vaccine acts in bi-stable dynamics when chaotic and periodic attractors coexist. We identify that this kind of vaccination acts as a control by destroying almost all the periodic basins. We explain this by the fact that chaotic attractors exhibit more desirable characteristics for epidemics than periodic ones in a bi-stable state.

Impact of periodic vaccination in SEIRS seasonal model.

We study three different strategies of vaccination in an SEIRS (Susceptible-Exposed-Infected-Recovered-Susceptible) seasonal forced model, which are (i) continuous vaccination; (ii) periodic short-time localized vaccination, and (iii) periodic pulsed width campaign. Considering the first strategy, we obtain an expression for the basic reproduction number and infer a minimum vaccination rate necessary to ensure the stability of the disease-free equilibrium (DFE) solution. In the second strategy, short duration pulses are added to a constant baseline vaccination rate. The pulse is applied according to the seasonal forcing phases. The best outcome is obtained by locating intensive immunization at inflection of the transmissivity curve. Therefore, a vaccination rate of 44.4% of susceptible individuals is enough to ensure DFE. For the third vaccination proposal, additionally to the amplitude, the pulses have a prolonged time width. We obtain a non-linear relationship between vaccination rates and the duration of the campaign. Our simulations show that the baseline rates, as well as the pulse duration, can substantially improve the vaccination campaign effectiveness. These findings are in agreement with our analytical expression. We show a relationship between the vaccination parameters and the accumulated number of infected individuals, over the years, and show the relevance of the immunization campaign annual reaching for controlling the infection spreading. Regarding the dynamical behavior of the model, our simulations show that chaotic and periodic solutions as well as bi-stable regions depend on the vaccination parameters range.

Selection of models and parameter estimation for monitoring the COVID-19 epidemic in Brazil via Bayesian inference

In 2019, a new strain of coronavirus led to an outbreak of disease cases named COVID-19, evolving rapidly into a pandemic. In Brazil, delayed decision making and lack of knowledge have resulted in an alarming increase in daily transmission and deaths. In this context, researchers used mathematical models to assist in determining the parameters that act in the spread of diseases, revealing containment measures. However, numerous mathematical models exist in the literature, each with specific parameters to be specified, leading to an important question about which model best represents the pandemic behavior. In this regard, this work aims to apply the Approximate Bayesian Computation method to select the best model and simultaneously estimate the parameters to resolve the abovementioned issue. The models adopted were susceptible-infected-recovered (SIR), susceptible-exposed-infected-recovered (SEIR), susceptible-infected-recovered-susceptible (SIRS), and susceptible-exposed-infected-recovered-susceptible (SEIRS). Approximate Bayesian Computation Monte Carlo Sequencing (ABC-SMC) was used to select among four competing models to represent the number of infected individuals and to estimate the model parameters based on three periods of Brazil COVID-19 data. A forecasting test was performed to test the ABC-SMC algorithm and the selected models for two months. The result was compared with the actual number of infected that were reported. Among the teste models, it was found that the ABC-SMC algorithm had a promising performance, since the data were noisy and the models could not predict all parameters.

A Method for Simulating the Propagation of Network Hot Topics Based on Combined SEIRS-ARIMA Model

The spread of hot topics on the internet can easily attract public attention and discussion. However, the spread of rumours can cause social instability, while positive media guidance can generate economic value. Therefore, developing a model that can understand the propagation of popular topics on the internet is a worthwhile pursuit. In this study, a hybrid model is developed by combining the SEIRS (Susceptible-Exposed-Infected-Recovered-Susceptible) infectious disease model to simulate the overall trend and the ARIMA (Autoregressive Integrated Moving Average) model to simulate noise, to explain internet hot topic propagation. Using the wordle game as a case study, the model exhibits a notable capability in conforming to the general pattern and the majority of the minor variations, while demonstrating a high degree of interpretability.

Model Matematika SEIRS Penyebaran Penyakit Pneumonia pada Balita dengan Pengaruh Vaksinasi

This research aims to establish a model for the spread of pneumonia in infants with SEIRS (Susceptible-Exposed-Infected-Recovered-Susceptible) type, analyze the model, and determine the minimum proportion of vaccinations. This model is divided into four classes, namely susceptible, infected but not yet active, infected, and cured. The stability analysis of the model was carried out disease-free and endemic by showing the stability of asymptotes in both, meaning that the disease would not disappear but with complete vaccine administration it could reduce the number of pneumonia spreads. The basic reproduction number for disease-free and endemic simulations is greater than 1 if the vaccine is incomplete and less than 1 if the vaccine is given completely

Mathematical model for transmission of Chlamydia due to sexual activity and unhygienic environment

Aim: Sexually transmitted diseases (STDs) need to be studied systematically to better understand their global spread. Transmission of Chlamydia trachomatis is a severe public health issue, with roughly 90 million new cases per year. Globally, Chlamydia trachomatis is the most frequent bacterial cause of STDs. Methods: To better understand the dynamics and transmission of Chlamydia, the susceptible-exposed-infected-recovered-susceptible (SEIRS) model was constructed. Using a system of nonlinear ordinary differential equations, a basic reproduction number has been calculated at an equilibrium point, and the system is locally and globally asymptotically stable at both disease-free and endemic equilibrium points. Numerical simulations illustrate the behavior and flow of Chlamydia infections in different compartments. Results: Conclude from the proposed study that 25% of individuals have been exposed to Chlamydia, of which 20% of individuals get infections due to sexual activity and 55% of individuals get recovered. 20% of individuals have been exposed to Chlamydia, of which 37% of individuals have been infected due to an unhygienic environment. Of those, 43% of individuals recovered. Also, it has been found that people are more likely to get infections because of an unhygienic environment than sexually active people. The recovery rate is also much better for people who have been infected because of an unhygienic environment. Conclusions: Sexually transmitted infections can be reduced by up to 10%. While infection due to an unhygienic environment can be controlled up to a certain intensity. According to this research, public awareness campaigns and the improvement of personal hygiene will play a major role in reducing the spread of the epidemic in the future.

QUALITATIVE ANALYSIS OF SEIRS ENDEMIC MODEL BOTH FROM PDEs AND ODEs PERSPECTIVE

An age-structured susceptible–exposed–infected–recovered–susceptible (SEIRS) endemic model is proposed in this analysis utilizing the tools of partial differential equations. Because of different outflows and inflows that are lopsided by migration and demographics factors, the population is supposed to be not constant. To demonstrate that the model is well-posed, an abstract Cauchy problem is developed from the proposed system. The simple reproduction number [Formula: see text] is used to analyze the local and global behavior of the disease-free equilibrium. The disease present equilibrium point is shown to exist and be stable locally under appropriate assumptions and conditions. We consider the age-free parameters and the problem is converted into an ordinary differential equations (ODEs) model. The ODEs model is investigated for disease-free and endemic equilibria and the global stability of each equilibrium is presented therein. A few simulations are carried out and discussed at the end of the paper to explain the central theorem of the study.

A Susceptible Exposed Infected Recovered Susceptible (SEIRS) Model for the Transmission of Tuberculosis

In this paper, a deterministic mathematical model was proposed and analyzed to understand the dynamics of tuberculosis based on the SEIRS model. The disease-free equilibrium, the endemic equilibrium, and their stabilities were examined. The R0 (basic reproduction number) was derived using the Next Generation Matrix method and its sensitivity analysis showed that the birth rate and infectious rate were the most sensitive parameters of R0. The behaviour of exposed individuals at the latent period with varied treatment rates were examined through numerical simulation. From the analysis carried out, the effect of variations of the treatments of latent TB shows that it affects the disease burden. This implies that testing and treatment of latent TB are important in preventing it from becoming infectious. The re-infection rate was examined to see the effect it had both on the recovered and susceptible populations. The study concludes by recommending the extension of the model to an age structured model with co-infection with another respiratory infectious disease like COVID-19.
 Keywords: Epidemiology; Latent TB treatment; Basic Reproduction Number; sensitivity analysis; numerical simulation

SARS-CoV-2 infection dynamics in Denmark, February through October 2020: Nature of the past epidemic and how it may develop in the future.

Initially, the relative sizes of the asymptomatic and the symptomatic infected populations were not known for the COVID-19 pandemic and neither was the actual fatality rate. Therefore it was not clear either how the pandemic would impact the healthcare system. As a result it was initially predicted that the COVID-19 epidemic in Denmark would overwhelm the healthcare system and thus both the diagnosis and treatment of other hospital patients were compromised for an extended period. To develop a mathematical model, which includes both asymptomatic and symptomatic infected persons, for early estimation of the epidemic's course, its Infection Fatality Rate and the healthcare system load in Denmark, both retrospectively and prospectively. The SEIRS (Susceptible-Exposed-Infected-Recovered-Susceptible) model including deaths outside hospitals and separate assessments of symptomatic and asymptomatic cases (based on seroprevalence) with different immunological memories. Optimal model parameters are in part identified by Monte Carlo based Least Square Error methods while micro-outbreaks are modeled by noise and explored in Monte Carlo simulations. Estimates for infected population sizes are obtained by using a quasi steady state method. The calculations and simulations made by the model were shown to fit with the observed development of the COVID-19 epidemic in Denmark. The antibody prevalence in the general population in May 2020 was 1.37%, which yields a relative frequency of symptomatic and asymptomatic cases of 1 to 5.2. Due to the large asymptomatic population, the Infection Mortality Rate was only 0.4%. However, with no non-pharmacological restrictions the COVID-19 death toll was calculated to have more than doubled the national average yearly deaths within a year. The transmission rate ℜ0 was 5.4 in the initial free epidemic period, 0.4 in the lock-down period and 0.8-1.0 in the successive re-opening periods through August 2020. The large asymptomatic population made the termination of the epidemic difficult and micro-outbreaks occurred when the country re-opened. The estimated infected population size July 15 to August 15 was 2,100 and 12,200 for October 1-20, 2020. The results of the model show, that COVID-19 has a low Infection Fatality Rate because the majority of infected persons are either asymptomatic or with few symptoms. A minority of the infected persons, therefore, requires hospitalization. That means that for a given infection pressure of both symptomatic and asymptomatic infected there will be a lower pressure on the capacity of the health care system than previously predicted. Further the epidemic will be difficult to terminate since about 84% of the infected individuals are asymptomatic but still contagious. The model may be useful if a major infection wave occurs in the autumn-winter season as it could make robust estimates both for the scale of an ongoing expanding epidemic and for the expected load on the healthcare system. The simulation may also be useful to evaluate different testing strategies based on estimated infected population sizes. The model can be adjusted and scaled to other regions and countries, which is illustrated with Spain and USA.

Mathematical Epidemiology Model Analysis on the Dynamics of COVID-19 Pandemic

In the present work, Susceptible-Exposed-Infected-Recovered-Susceptible (SEIRS) mathematical model for COVID-19 Pandemic is formulated and analyzed. The positivity, boundedness, and existence of the solutions of the model are proved. The Disease-free equilibrium point and endemic equilibrium points are identified. Local Stability of disease-free Equilibrium point is checked with the help of Next generation matrix. Global stability of endemic equilibrium point is proved using the Concept of Liapunove function. The basic reproduction number for Novel Corona virus pandemic is computed as R₀ = (αβΛ) ⁄ [(δ + μ) (β + δ + μ) (γ + δ + μ)] which depend on six different parameters. It is observed that if basic reproduction number is less than one, then number of cases decrease over time and eventually the disease dies out, and if the basic reproduction number is equals to one, then number of cases are stable. On the other hand, if the basic reproduction number is greater than one, then the number of cases increase over time gets worth. Sensitivity analysis of the basic reproduction number is done with respect to each parameter. It is observed that only some parameters Λ, α, β have high impact on the basic reproduction number. Consequently, with real data on the parameter it is helpful to predict the disease persistence or decline in the present situation. Lastly, numerical simulations are given using DEDiscover software to illustrate analytical results.

A spatial SEIRS reaction-diffusion model in heterogeneous environment

We propose a susceptible-exposed-infected-recovered-susceptible (SEIRS) reaction-diffusion model, where the disease transmission and recovery rates can be spatially heterogeneous. The basic reproduction number (R0) is connected with the principal eigenvalue of a linear cooperative elliptic system. Threshold-type results on the global dynamics in terms of R0 are established. The monotonicity of R0 with respect to the diffusion rates of the exposed and infected individuals, which does not hold in general, is established in several cases. Finally, the asymptotic profile of the endemic equilibrium is investigated when the diffusion rate of the susceptible individuals is small. Our results reveal the importance of the movement of the exposed and recovered individuals in disease dynamics, as opposed to most of previous works which solely focused on the movement of the susceptible and infected individuals.

Modeling multi-group dynamics of related viral videos with delay differential equations

Epidemic models have previously been employed to better understand the spread of internet content, such as individual viral videos. Motivated by geographical differences between viewing populations, as well as by multiple viral videos which are linked or related in some way, we propose a multi-group susceptible–exposed–infected–recovered–susceptible (SEIRS) delay differential equation epidemic model for the popularity evolution of viral videos. The model accounts for the multi-group environment via (in general, heterogeneous) networks of single-group SEIRS equations. Existence and stability conditions for disease-free and endemic equilibrium states are obtained, while numerical simulations demonstrate the emergence of non-symmetric endemic equilibrium states across the network — in the case of heterogeneous parameters. In order to demonstrate the effectiveness of our modeling approach, we compare solutions to our model with data obtained from coupled YouTube videos. Our results demonstrate that the inclusion of coupling between such videos allows for better agreement with data than considering each video in isolation. Such results suggest that the multi-group epidemic modeling approach including time delays and coupling between videos is a useful tool for understanding the dynamics of viral videos which are connected in some manner. While the proposed network SEIRS differential delay equation epidemic model was applied to viral video data, it may be applicable to a wide variety of internet content and, more generally, may be used to model information propagation within and between communities under the assumption of time delays in information propagation.

Understanding viral video dynamics through an epidemic modelling approach

Motivated by the hypothesis that the spread of viral videos is analogous to the spread of a disease epidemic, we formulate a novel susceptible–exposed–infected–recovered–susceptible (SEIRS) delay differential equation epidemic model to describe the popularity evolution of viral videos. Our models incorporate time-delay, in order to accurately describe the virtual contact process between individuals and the temporary immunity of individuals to videos after they have grown tired of watching them. We validate our models by fitting model parameters to viewing data from YouTube music videos, in order to demonstrate that the model solutions accurately reproduce real behaviour seen in this data. We use an SEIR model to describe the initial growth and decline of daily views, and an SEIRS model to describe the long term behaviour of the popularity of music videos. We also analyse the decay rates in the daily views of videos, determining whether they follow a power law or exponential distribution. Although we focus on viral videos, the modelling approach may be used to understand dynamics emergent from other areas of science which aim to describe consumer behaviour.

Malware propagation modeling considering software diversity and immunization

In this paper, we propose a discrete-time susceptible–exposed–infected–recovered–susceptible (SEIRS) epidemic model of malware propagation in scale-free networks (SFNs) with considering software diversity. We study dynamical behavior of the SEIRS model, which is determined by a threshold (i.e., basic reproductive ratio). With this threshold, we can predict whether the malware propagates or not. Also, using a coloring algorithm, the number of diverse software packages installed on nodes is calculated and used as a parameter to prevent malware spreading. Furthermore, we investigate global dynamics of the model and analyze the stability of the malware-free equilibrium. The dynamics of malware propagation is evaluated using the results of numerical simulations. Simulation results show that the proposed model, which considers software diversity, is more effective than other existing epidemic models. We have compared different immunization mechanisms, and have shown that the targeted immunization is better than the random immunization for controlling malware spreading in SFNs.

USEIRS model for the contagion of individual aggressive behavior under emergencies

Individual aggressive behavior under emergencies is contagious, and often leads to collective aggressive behavior. In this paper, we apply the epidemic model to investigate the contagion of individual aggressive behavior under emergencies, extending the SEIRS (Susceptible–Exposed–Infected–Recovered–Susceptible) model by adding a new group of people – uninducible individuals. Thus, a new dynamic model USEIRS (Uninducible–Susceptible–Exposed–Infected–Recovered–Susceptible) is developed. The threshold of individual aggressive behavior contagion is deduced from the USEIRS model through the analysis of the eliminating and prevailing stabilities and equilibrium of aggressive behavior contagion. The numerical solutions of the USEIRS model show that a higher number of initial uninducible individuals can reduce the number of people with aggressive behavior. However, the decrease in the number of aggressive individuals will be insignificant if the uninducible individuals have little influence on the public. A higher uninducible rate can reduce the number of individuals with aggressive behavior. However, some people will still inevitably behave aggressively at the beginning. The effect of higher uninducible rate has an accelerating feature, which becomes more obvious with the development of emergency. Providing information and education to increase the uninducible population, or more communication between experts, government officials and the general public to increase the uninducible rate are strategies for reducing individual aggressive behavior.

Transmission dynamics of Chlamydia trachomatis affect the impact of screening programmes

To assess the impact of screening programmes in reducing the prevalence of Chlamydia trachomatis, mathematical and computational models are used as a guideline for decision support. Unfortunately, large uncertainties exist about the parameters that determine the transmission dynamics of C. trachomatis. Here, we use a SEIRS (susceptible-exposed-infected-recovered-susceptible) model to critically analyze the turnover of C. trachomatis in a population and the impact of a screening programme. We perform a sensitivity analysis on the most important steps during an infection with C. trachomatis. Varying the fraction of the infections becoming symptomatic as well as the duration of the symptomatic period within the range of previously used parameter estimates has little effect on the transmission dynamics. However, uncertainties in the duration of temporary immunity and the asymptomatic period can result in large differences in the predicted impact of a screening programme. We therefore analyze previously published data on the persistence of asymptomatic C. trachomatis infection in women and estimate the mean duration of the asymptomatic period to be longer than anticipated so far, namely 433days (95% CI: 420–447days). Our study shows that a longer duration of the asymptomatic period results in a more pronounced impact of a screening programme. However, due to the slower turnover of the infection, a substantial reduction in prevalence can only be achieved after screening for several years or decades.