Stochastic Epidemic Model Research Articles

Stationary Distribution and Density Function for a High-Dimensional Stochastic SIS Epidemic Model with Mean-Reverting Stochastic Process

This paper explores a high-dimensional stochastic SIS epidemic model characterized by a mean-reverting, stochastic process. Firstly, we establish the existence and uniqueness of a global solution to the stochastic system. Additionally, by constructing a series of appropriate Lyapunov functions, we confirm the presence of a stationary distribution of the solution under R0s>1. Taking 3D as an example, we analyze the local stability of the endemic equilibrium in the stochastic SIS epidemic model. We introduce a quasi-endemic equilibrium associated with the endemic equilibrium of the deterministic system. The exact probability density function around the quasi-stable equilibrium is determined by solving the corresponding Fokker–Planck equation. Finally, we conduct several numerical simulations and parameter analyses to demonstrate the theoretical findings and elucidate the impact of stochastic perturbations on disease transmission.

Analysis of stochastic epidemic model with awareness decay and heterogeneous individuals on multi-weighted networks.

In the current study, we present a stochastic SAIS (unaware susceptible-aware susceptible-infectious-unaware susceptible) epidemic dynamic model on complex networks with multi-weights. The disease dynamic is influenced by random perturbations to the force of the infection rates, as well as awareness rates. To analyze the problem of extinction, we discuss both the stochastic asymptotic stability in the large and almost surely exponential stability of the trivial solution. Then, we get some sufficient conditions, which guarantee the stochastic persistence of infectious disease. Based on the Erdös-Réyni random graph, the numerical simulations are given. These not only validate our conclusions but also obtain else significative results. Both theoretical results and numerical simulations further reflect that improvement of risk awareness and reduction of decay in awareness are highly effective in preventing disease spread. And then, environmental noises play a significant role in disease transmission.

Dynamical behaviors of a stochastic SIVS epidemic model with the Ornstein-Uhlenbeck process and vaccination of newborns.

In this paper, we study a stochastic SIVS infectious disease model with the Ornstein-Uhlenbeck process and newborns with vaccination. First, we demonstrate the theoretical existence of a unique global positive solution in accordance with this model. Second, adequate conditions are inferred for the infectious disease to die out and persist. Then, by classic Lynapunov function method, the stochastic model is allowed to obtain the sufficient condition so that the stochastic model has a stationary distribution represents illness persistence in the absence of endemic equilibrium. Calculating the associated Fokker-Planck equations yields the precise expression of the probability density function for the linearized system surrounding the quasi-endemic equilibrium. In the end, the theoretical findings are shown by numerical simulations.

Intelligent Adaptive Networks for Chaos Analysis in Stochastic SIS Differential Epidemic Model with the Impact of Unpredictable Environmental Factors

In this study, an intelligent computing framework is presented to explore the influence of unpredictable environmental factors on the spread of infections. A stochastic non-linear SIS epidemic model (SISEM) is examined that incorporates a non-linear incidence rate, and impact of random fluctuations on the disease transmission. The knacks of artificial neural networks Levenberg–Marquardt (ANNLMA) technique are exploited to illustrate the behavioral complexities of stochastic non-linear SISEM, which explores the dynamics of disease transmission in a stochastic environment and provides insights into the interactions between susceptible and infectious populations. The Milstein approach of order 1 generates a dataset for the proposed ANNLMA technique by varying integrated parameters, such as the rates of recovery, death, disease transmission, noise standard deviation and the entire population, for various SISEM scenarios. The referenced dataset is randomly divided into test, validation and train sets for the weight learning of the proposed ANNLMA. The viability of the ANNLMA is measured using intensive simulation-based investigations through analysis of mean squared errors, histograms, regression and autocorrelation studies. The investigations highlighted that the ANNLMA results closely match the reference results, illustrating convergence within the 10[Formula: see text] to 10[Formula: see text] range, indicating the validity of the proposed methodology.

A stochastic SIRS epidemic model with a saturation incidence under semi-Markovian switching

A stochastic SIRS epidemic model with a saturation incidence under semi-Markovian switching

Lévy noise impact on a stochastic epidemic model with healing, relapse and vaccination

In this work, we propose and analyze a stochastic Susceptible-Infected-Recovered-Healed (SIRH) epidemic model with healing, relapse, and jump effects to investigate the impact of vaccination on the disease dynamics. The stochastic perturbations are introduced into the model to account for the inherent randomness in the disease transmission and recovery processes. The existence and uniqueness of the positive global solution for the stochastic model are established. Further, we study the conditions for the extinction and persistence of the disease. The impact of vaccination is incorporated into the model by considering the vaccination rate as a control parameter. Numerical simulations are carried out to illustrate the dynamical behavior of the stochastic SIRH model and to analyze the effects of vaccination on the disease spread. The graphical representations of the approximate solutions for the susceptible, infected, recovered, and healed populations are provided under different vaccination scenarios.

Instabilities and Pattern Formation in Epidemic Spread Induced by Nonlinear Saturation Effects and Ornstein–Uhlenbeck Noise

Abstract We analytically study the emergence of instabilities and the consequent steady-state pattern formation in a stochastic partial differential equation (PDE) based, compartmental model of spatiotemporal epidemic spread. The model is characterized by: (1) strongly nonlinear forces representing the infection transmission mechanism and (2) random environmental forces represented by the Ornstein–Uhlenbeck (O–U) stochastic process which better approximates real-world uncertainties. Employing second-order perturbation analysis and computing the local Lyapunov exponent, we find the emergence of diffusion-induced instabilities and analyze the effects of O–U noise on these instabilities. We obtain a range of values of the diffusion coefficient and correlation time in parameter space that support the onset of instabilities. Notably, the stability and pattern formation results depend critically on the correlation time of the O–U stochastic process; specifically, we obtain lower values of steady-state infection density for higher correlation times. Also, for lower correlation times the results approach those obtained in the white noise case. The analytical results are valid for lower-order correlation times. In summary, the results provide insights into the onset of noise-induced, and Turing-type instabilities in a stochastic PDE epidemic model in the presence of strongly nonlinear deterministic infection forces and stochastic environmental forces represented by Ornstein–Uhlenbeck noise.

Randomness suppress backward bifurcation in an epidemic model with limited medical resources

This paper delves into the dynamic features of a stochastic SIR epidemic model featuring a perturbed transmission rate influenced by white noise. Our primary aim is to unravel the intricate interplay between restricted medical resources, their supply efficiency, and environmental stochasticity, shedding light on their collective impact on the transmission dynamics of infectious diseases. Our findings bring to light a notable distinction from the deterministic counterpart of the model. Specifically, under varying scenarios of medical resource availability and supply efficiency, the stochastic model exhibits a departure from bifurcation phenomena. This stands in contrast to the deterministic model, which is characterized by the presence of both backward bifurcation and Hopf bifurcation phenomena. To complement and validate our theoretical findings, numerical simulations are employed, providing concrete illustrations of the dynamical phenomena discussed in the paper. This research contributes to a nuanced understanding of the intricate interplay between stochastic environmental factors, medical resource constraints, and disease transmission dynamics, offering valuable insights for public health management and epidemic control strategies.

Extinction Dynamics and Equilibrium Patterns in Stochastic Epidemic Model for Norovirus: Role of Temporal Immunity and Generalized Incidence Rates

Norovirus is a leading global cause of viral gastroenteritis, significantly affecting mortality, morbidity, and healthcare costs. This paper develops and analyzes a stochastic SEIQR epidemic model for norovirus dynamics, incorporating temporal immunity and a generalized incidence rate. The model is proven to have a unique positive global solution, with extinction conditions explored. Using Khasminskii’s method, the model’s ergodicity and equilibrium distribution are investigated, demonstrating a unique ergodic stationary distribution when R^s>1. Extinction occurs when R0E<1. Computer simulations confirm that noise level significantly influences epidemic spread.

Stationary distribution of a stochastic generalized SIRI epidemic model with reinfection and relapse

In this paper, we propose and investigate the stochastic SIRI epidemic model with two generalized incidence rate functions. We firstly study the existence and uniqueness of the globally positive solution to the stochastic SIRI model with positive initial value. Then we obtain sufficient conditions for the extinction of the disease in the stochastic epidemic model, and find that the large noise can make the disease die out exponentially. Meanwhile, we obtain that the solution to the stochastic model has a unique stationary distribution when R0s˜ is greater than one. Our results show that the intensity of white noise can affect the dynamical behaviors of the model. Finally, we use numerical simulation to illustrate theoretical results, and apply both the stochastic and deterministic models to analyze the outbreak of COVID-19 epidemic in Serbia.

Dynamics analysis of a delayed stochastic SIRS epidemic model with a nonlinear incidence rate

. This article proposes a stochastic delayed SIRS epidemic model with nonlinear incidence to describe disease transmission in a stochastic environment. First, it proves that the model has a unique global positive solution and derives sufficient conditions for the extinction and persistence of the epidemic in the population under appropriate conditions. Then, by constructing suitable Lyapunov functions, the asymptotic behavior of the model around the disease-free equilibrium and endemic equilibrium points of the deterministic model is analyzed, and it is concluded that under certain conditions, the solution of the stochastic system randomly oscillates around these two equilibrium points. Finally, several numerical examples are provided to validate the theoretical results, illustrating the significant impact of stochastic perturbations and time delays on disease control.

Long-time behavior and quasi-density function for a stochastic epidemic model with relapse and reinfection

A stochastic susceptible–infected–recovered–infected (SIRI) epidemic model with relapse and reinfection is established in this paper. First, we prove that the solution to the epidemic model is unique and globally positive. Next, we determine some sufficient conditions for the extinction of the disease when [Formula: see text] and for the persistence in mean in the case of [Formula: see text]. Furthermore, we prove the existence of at least one ergodic stationary distribution of the stochastic model if [Formula: see text]. Additionally, by solving the corresponding three-dimensional Fokker–Planck equation, it is theoretically shown that the epidemic model has a log-normal probability density function when [Formula: see text], then we obtain the exact expression of density function of the stationary distribution. Finally, we give some numerical simulations to support our theoretical results.

Global insights into a stochastic SIRS epidemic model with Beddington–DeAngelis incidence rate

Global insights into a stochastic SIRS epidemic model with Beddington–DeAngelis incidence rate

An unconditional boundary and dynamics preserving scheme for the stochastic epidemic model

An unconditional boundary and dynamics preserving scheme for the stochastic epidemic model

Domain preserving and strongly converging explicit scheme for the stochastic SIS epidemic model

In this article, we construct a numerical method for a stochastic version of the Susceptible–Infected–Susceptible (SIS) epidemic model, expressed by a suitable stochastic differential equation (SDE), by using the semi-discrete method to a suitable transformed process. We prove the strong convergence of the proposed method, with order 1, and examine its stability properties. Since SDEs generally lack analytical solutions, numerical techniques are commonly employed. Hence, the research will seek numerical solutions for existing stochastic models by constructing suitable numerical schemes and comparing them with other schemes. The objective is to achieve a qualitative and efficient approach to solving the equations. Additionally, for models that have not yet been proposed for stochastic modeling using SDEs, the research will formulate them appropriately, conduct theoretical analysis of the model properties, and subsequently solve the corresponding SDEs.

Dynamic analysis of a stochastic SVEIR epidemic model with saturated incidence and partial vaccination

Dynamic analysis of a stochastic SVEIR epidemic model with saturated incidence and partial vaccination

Stochastic EM algorithm for partially observed stochastic epidemics with individual heterogeneity.

We develop a stochastic epidemic model progressing over dynamic networks, where infection rates are heterogeneous and may vary with individual-level covariates. The joint dynamics are modeled as a continuous-time Markov chain such that disease transmission is constrained by the contact network structure, and network evolution is in turn influenced by individual disease statuses. To accommodate partial epidemic observations commonly seen in real-world data, we propose a stochastic EM algorithm for inference, introducing key innovations that include efficient conditional samplers for imputing missing infection and recovery times which respect the dynamic contact network. Experiments on both synthetic and real datasets demonstrate that our inference method can accurately and efficiently recover model parameters and provide valuable insight at the presence of unobserved disease episodes in epidemic data.

Dynamics of a multigroup stochastic SIQR epidemic model

Abstract In this paper, we consider a multigroup stochastic SIQR epidemic model with varying total population size. After proving the existence and uniqueness of the global solution to the system, we developed sufficient conditions for the existence of an stationary ergodic distribution of the positive solutions. Then we gave sufficient conditions for extinction of the diseases which is based on the basic reproduction number in its corresponding deterministic system.

Analysis of a stochastic SEIuIrR epidemic model incorporating the Ornstein-Uhlenbeck process

This article aims to analyze a stochastic epidemic model SEIuIrR (Susceptible-exposed-undetected infected-detected infected (reported -recovered) assuming that the transmission rate at which people undetected become detected is perturbed by the Ornstein–Uhlenbeck process. Our first objective is to prove that the stochastic model has a unique positive global solution by constructing a nonnegative Lyapunov function. Afterward, we provide a sufficient criterion to prove the existence of an ergodic stationary distribution of the mode by constructing a suitable series of Lyapunov functions. Subsequently, we establish sufficient conditions for the extinction of the disease. Finally, a series of numerical simulations are carried out to illustrate the theoretical results.

Extinction and stationary distribution of a novel SIRS epidemic model with general incidence rate and Ornstein–Uhlenbeck process

We propose, in this paper, a novel stochastic SIRS epidemic model to characterize the effect of uncertainty on the distribution of infectious disease, where the general incidence rate and Ornstein–Uhlenbeck process are also introduced to describe the complexity of disease transmission. First, the existence and uniqueness of the global nonnegative solution of our model is obtained, which is the basis for the discussion of the dynamical behavior of the model. And then, we derive a sufficient condition for exponential extinction of infectious diseases. Furthermore, through constructing a Lyapunov function and using Fatou’s lemma, we obtain a sufficient criterion for the existence and ergodicity of a stationary distribution, which implies the persistence of the disease. In addition, the specific form of the density function of the model near the quasiendemic equilibrium is proposed by solving the corresponding Fokker–Planck equation and using some relevant algebraic equation theory. Finally, we explain the above theoretical results through some numerical simulations.