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The complex interactions were performed among non-pharmaceutical interventions, vaccinations, and hosts for all epidemics in mainland China during the spread of COVID-19. Specially, the small-scale epidemic in the city described by SVEIR model was less found in the current studies. The SVEIR model with control was established to analyze the dynamical and epidemiological features of two epidemics in Jinzhou City led by Omicron variants before and after Twenty Measures. In this study, the total population (N) of Jinzhou City was divided into five compartments: the susceptible (S), the vaccinated (V), the exposed (E), the infected (I), and the recovered (R). By surveillance data and the SVEIR model, three methods (maximum likelihood method, exponential growth rate method, next generation matrix method) were governed to estimate basic reproduction number, and the results showed that an increasing tendency of basic reproduction number from Omicron BA.5.2 to Omicron BA.2.12.1. Meanwhile, the effective reproduction number for two epidemics were investigated by surveillance data, and the results showed that Jinzhou wave 1 reached the peak on November 1 and was controlled 7 days later, and that Jinzhou wave 2 reached the peak on November 28 and was controlled 5 days later. Moreover, the impacts of non-pharmaceutical interventions (awareness delay, peak delay, control intensity) were discussed extensively, the variations of infection scales for Omicron variant and EG.5 variant were also discussed. Furthermore, the investigations on peaks and infection scales for two epidemics in dynamic zero-COVID policy were operated by the SVEIR model with control. The investigations on public medical requirements of Jinzhou City and Liaoning Province were analyzed by using SVEIR model without control, which provided a possible perspective on variant evolution in the future.

The epidemiological characteristics and distributions of two epidemics in Fujian Province of Southeast China were attributed to the complex interactions among variant, host, and non-pharmaceutical interventions (NPIs). All reported cases in the Putian epidemic (September 8–October 2, 2021, Delta variant B.1.617.2) and Fuzhou epidemic (October 22–November 18, 2022, Omicron variant BA.5.2) were classified by sex, age group, occupation, and location in this study. Using surveillance data from the Fujian Provincial Center for Disease Control and Prevention, we established a virus-oriented SVEIR (Susceptible–Vaccinated–Exposed–Infected–Recovered) model to investigate the dynamic evolution features of these two variants and the effects of NPIs. The optimal simulations were carried out with variants and scenario investigations. The scenario investigations showed that NPIs significantly reduced the transmission risk and infection scales of COVID-19, and that the Omicron variant was more infectious than the Delta variant. Moreover, the dynamic investigations revealed the increasing tendencies from Delta to Omicron, such as the basic reproduction number, infection rate, percentage of high-risk cases, and the growth rate. Decreasing tendencies were also identified, such as the average recovery period, the awareness delay, and the percentage of symptomatic cases. This study highlighted that NPIs played critical roles in successfully containing the two epidemics. Such interventions are strongly recommended to public health policymakers.

In the entire world, pneumonia is one of the leading causes of death, which is particularly dangerous for young children (those under five years old) and the elderly (those over 65). A deterministic susceptible, vaccinated, exposed, infected, and recovered (SVEIR) model is used in this work to mathematically study the dynamics of pneumonia disease and examine stability analysis, basic reproduction numbers, and equilibrium points of dynamical systems theory models. Spatial equilibria are studied to model disease-free equilibria that are locally asymptotic stable. Numerical simulations of the model have been carried out using MATLAB21. The SVEIR flow and its variables for different parameter sets have been studied through numerical simulations. The solution to the issue is provided through the use of illustrated and explicated results. According to research findings, if vaccination rates rise over the necessary vaccination ratio, the sickness will finally vanish from the community. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with a diffusive SVEIR epidemic model with nonlinear incidences. We first obtain the well-posedness of solutions for the model. Then, the basic reproduction number R 0 and the local basic reproduction number R ¯ 0 (x) are calculated, which are defined as the spectral radii of the next-generation operators. The relationship between R 0 and R ¯ 0 (x) as well as the asymptotic properties of R 0 when the diffusive rates tend to infinity or zero is investigated by introducing two compact linear operators L 1 and L 2 . Using the theory of monotone dynamical systems and the persistence theory of dynamical systems, we show that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , while the disease is uniformly persistent when R 0 > 1 . Furthermore, in the spatially homogeneous case, by using the Lyapunov functions method and LaSalle's invariance principle, we completely obtain that the disease-free equilibrium is globally asymptotically stable if R 0 ≤ 1 , and the endemic equilibrium is globally asymptotically stable if R 0 > 1 and an additional condition is satisfied. [ABSTRACT FROM AUTHOR]

The effective reproduction number, $ R_t $, is a vital epidemic parameter utilized to judge whether an epidemic is shrinking, growing, or holding steady. The main goal of this paper is to estimate the combined $ R_t $ and time-dependent vaccination rate for COVID-19 in the USA and India after the vaccination campaign started. Accounting for the impact of vaccination into a discrete-time stochastic augmented SVEIR (Susceptible-Vaccinated-Exposed-Infectious-Recovered) model, we estimate the time-dependent effective reproduction number $ (R_t) $ and vaccination rate $ (\xi_t) $ for COVID-19 by using a low pass filter and the Extended Kalman Filter (EKF) approach for the period February 15, 2021 to August 22, 2022 in India and December 13, 2020 to August 16, 2022 in the USA. The estimated $ R_t $ and $ \xi_t $ show spikes and serrations with the data. Our forecasting scenario represents the situation by December 31, 2022 that the new daily cases and deaths are decreasing for the USA and India. We also noticed that for the current vaccination rate, $ R_t $ would remain greater than one by December 31, 2022. Our results are beneficial for the policymakers to track the status of the effective reproduction number, whether it is greater or less than one. As restrictions in these countries ease, it is still important to maintain safety and preventive measures.

Based on the spread of COVID-19, in the present paper, an imperfectly vaccinated SVEIR model for latent age is proposed. At first, the equilibrium points and the basic reproduction number of the model are calculated. Then, we discuss the asymptotic smoothness and uniform persistence of the semiflow generated by the solutions of the system and the existence of an attractor. Moreover, LaSalle's invariance principle and Volterra type Lyapunov functions are used to prove the global asymptotic stability of both the disease-free equilibrium and the endemic equilibrium of the model. The conclusion is that if the basic reproduction number R ρ is less than one, the disease will gradually disappear. However, if the number is greater than one, the disease will become endemic and persist. In addition, numerical simulations are also carried out to verify the result. Finally, suggestions are made on the measures to control the ongoing transmission of COVID-19. [ABSTRACT FROM AUTHOR]

Measles is a highly contagious viral disease that can lead to serious complications, including death, particularly in young children. In this study, we developed a mathematical model that incorporates a seasonal transmission parameter to examine the measles transmission dynamics. We define the basic reproduction number ( R 0 ) and show its utility as a threshold parameter for global dynamics and the existence of periodic solutions. The model was applied to the measles outbreak that occurred in Pakistan from 2019 to 2021 and provided a good fit to the observed data. Our estimate of the basic reproduction number was found to be greater than one, indicating that the disease will persist in the population. The findings highlight the need to increase vaccination coverage and efficacy to mitigate the impact of the epidemic. The model also shows the long-term behavior of the disease, which becomes endemic and recurs annually. Our simulations demonstrate that a shorter incubation period accelerates the spread of the disease, while a higher vaccination coverage rate reduces its impact. The importance of the second dose of the measles vaccine is emphasized, and a higher vaccine efficacy rate can also help bring R 0 below one. Our study provides valuable information for the development and implementation of effective measles control strategies. To prevent future outbreaks, increasing vaccination coverage among the population is the most effective way to reduce the transmission of measles. [ABSTRACT FROM AUTHOR]

An Susceptible-Vaccinated-Exposed-Infectious-Recovered computer virus model with nonlinear incidence rate and two delays is proposed and its Hopf bifurcation is investigated. The existence of Hopf bifurcation at the viral equilibrium is established by analyzing its distribution of characteristic values. Moreover, the stability and the direction of the Hopf bifurcation are obtained by using the normal form theory and centre manifold theory. Finally, a numerical simulation is performed to validate the theoretical results obtained in the paper.

Abstract We are concerned with the Hopf bifurcation of an SVEIR computer virus model with time delay and nonlinear incident rate. First of all, by analyzing the associated characteristic equation we obtain sufficient conditions for its local stability and the existence of a Hopf bifurcation. Directly afterward, by means of the normal form theory and the center manifold theorem we derive explicit formulas that determine the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions. Finally, we carry out numerical simulations to illustrate and verify the theoretical results.

An Susceptible-Vaccinated-Exposed-Infectious-Recovered computer virus model with nonlinear incidence rate and two delays is proposed and its Hopf bifurcation is investigated. The existence of Hopf bifurcation at the viral equilibrium is established by analyzing its distribution of characteristic values. Moreover, the stability and the direction of the Hopf bifurcation are obtained by using the normal form theory and centre manifold theory. Finally, a numerical simulation is performed to validate the theoretical results obtained in the paper. [ABSTRACT FROM AUTHOR]

This paper is devoted to Hopf bifurcation of a delayed SVEIR model with partial immunization that describes worms propagation on internet. Sufficient conditions for existence of Hopf bifurcation are obtained by considering the latent period time delay of worms as the bifurcation parameter. Properties of Hopf bifurcation are then investigated with the help of the normal form theory and the center manifold theorem. Numerical simulations show that worms propagation in internet can be controlled and eliminated by shortening the time delay.

Some studies show that the efficacy of vaccines may decline and the vaccine-induced immunity wanes with time elapsed since vaccination (e.g., BCG, HBV vaccines). Meanwhile, the use of media can effectively contain the emerging epidemics. In our work, an SVEIR epidemic model with media impact, age-dependent vaccination and latency is proposed, where the efficacy of vaccines depends on the time since vaccination. Uniform persistence and the sharp threshold dynamics are established, that is, the vaccination reproduction number completely determines the global dynamics of the model system. Our main results established can be extended to the SVEIR model with the general nonlinear infectious force function of monotonicity and concavity. [ABSTRACT FROM AUTHOR]

An SVEIR epidemic model with imperfect vaccination and nonlinear incidence, and a general latent distribution is formulated. By constructing Lyapunov functionals, it is shown that the disease will die out if the vaccination reproduction number R v a c ≤ 1 and the disease becomes endemic if R v a c > 1 . Furthermore, vaccination effects are analyzed. Two special forms the probability of remaining in latent class are discussed. When the probability is negatively exponentially distributed, we present an efficient approach of proving global stability of the endemic equilibrium of the SVEIR system of ordinary differential equations (ODEs), which may improve some known approaches. When the probability is a step-function, the delay differential equation (DDE) system derived is used to study the impacts of vaccination and saturated incidence on the mumps transmission. [ABSTRACT FROM AUTHOR]

The effective reproduction number, $ R&#95;t $, is a vital epidemic parameter utilized to judge whether an epidemic is shrinking, growing, or holding steady. The main goal of this paper is to estimate the combined $ R&#95;t $ and time-dependent vaccination rate for COVID-19 in the USA and India after the vaccination campaign started. Accounting for the impact of vaccination into a discrete-time stochastic augmented SVEIR (Susceptible-Vaccinated-Exposed-Infectious-Recovered) model, we estimate the time-dependent effective reproduction number $ (R&#95;t) $ and vaccination rate $ (\xi&#95;t) $ for COVID-19 by using a low pass filter and the Extended Kalman Filter (EKF) approach for the period February 15, 2021 to August 22, 2022 in India and December 13, 2020 to August 16, 2022 in the USA. The estimated $ R&#95;t $ and $ \xi&#95;t $ show spikes and serrations with the data. Our forecasting scenario represents the situation by December 31, 2022 that the new daily cases and deaths are decreasing for the USA and India. We also noticed that for the current vaccination rate, $ R&#95;t $ would remain greater than one by December 31, 2022. Our results are beneficial for the policymakers to track the status of the effective reproduction number, whether it is greater or less than one. As restrictions in these countries ease, it is still important to maintain safety and preventive measures.

The vaccination period (i.e.,  the time period retaining acquired immunity following vaccination) and the latent period are two important factors affecting disease dynamics. Classical compartmental disease models unrealistically assume that these periods are exponentially distributed. In this paper, these two ages are assumed to have arbitrary distributions that are represented by age-specific rates leaving the vaccinated and the exposed classes, resulting in an SVEIR model with population dynamics. We investigate the global behavior of this model, and derive its basic reproduction number. The basic reproduction number completely determines the global dynamics of the model, i.e., the disease-free equilibrium is globally asymptotically stable and the disease always dies out when the basic reproduction number is less than unity; whereas when the basic reproduction number is larger than unity, there exists a unique endemic equilibrium which is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists. The contributions of the vaccine-wane rate to the basic reproduction number and the level of the endemic equilibrium are displayed. [ABSTRACT FROM AUTHOR]

We are concerned with the Hopf bifurcation of an SVEIR computer virus model with time delay and nonlinear incident rate. First of all, by analyzing the associated characteristic equation we obtain sufficient conditions for its local stability and the existence of a Hopf bifurcation. Directly afterward, by means of the normal form theory and the center manifold theorem we derive explicit formulas that determine the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions. Finally, we carry out numerical simulations to illustrate and verify the theoretical results. [ABSTRACT FROM AUTHOR]