Stochastic Epidemic Model Research Articles

A regime-switching stochastic SIR epidemic model with a saturated incidence and limited medical resources

The stochastic switching SIR epidemic model with saturated incidence and limited medical treatment is investigated in this paper. By using Lyapunov methods and Itô formula, we first prove that the system has a unique global positive solution with any positive initial value. Then combining inequality technique and the ergodic property of Markov switching, the sufficient conditions for extinction and persistence in the mean of the disease are established. The results demonstrate that increasing medical resources and improving supply efficiency can accelerate the transition from the persistent state to the extinct state. Meanwhile, the high incidence rate will slow down the extinction of the disease. Specially, the switching noise can induce the system to toggle between the extinct and persistent states. Finally, some numerical simulations are carried out to confirm the analytical results.

Dynamics and stationary distribution of a stochastic SIRS epidemic model with a general incidence and immunity

Infected individuals often obtain or lose immunity after recovery in medical studies. To solve the problem, this paper proposes a stochastic SIRS epidemic model with a general incidence rate and partial immunity. Through an appropriate Lyapunov function, we obtain the existence and uniqueness of a unique globally positive solution. The disease will be extinct under the threshold criterion. We analyze the asymptotic behavior around the disease-free equilibrium of a deterministic SIRS model. By using the Khasminskii method, we prove the existence of a unique stationary distribution. Further, solutions of the stochastic model fluctuate around endemic equilibrium under certain conditions. Some numerical examples illustrate the theoretical results.

A multi‐group SEIRI epidemic model with logistic population growth under discrete Markov switching: Extinction, persistence, and positive recurrence

This paper is concerned with the stochastic dynamics of a multi‐group stochastic SEIRI epidemic model with logistic population growth, standard incidence rate, and Markovian switching. For this purpose, we first show that the solution of the stochastic system is positive and global. Then we obtain sufficient conditions for the extinction of disease. In addition, the persistence in the mean of disease is proved. Specifically, by constructing suitable stochastic Lyapunov functions, we find a domain that is positive recurrence for the solution of stochastic system. Finally, numerical simulations are employed to verify our analytical results.

Near-optimal control of a stochastic partial differential equation SEIR epidemic model under economic constraints

The outbreak of a disease has negative effects on the economy, and economic activities affect the spread of a disease. In this paper, based on the relationship between economic activities and infectious diseases, we establish a SEIRG (susceptible-exposure-infective-recovered-economic) model. Applying the near-maximum condition of a Hamiltonian function, we provide sufficient and necessary conditions for the near optimal control problem of the SEIRG system. In addition, we obtain the optimal control function of the research system. Finally, we develop an algorithm for a near optimal control problem and use numerical simulations to illustrate the effects of vaccination and treatment.

Applications of the Delay Stochastic Simulation Algorithm (DSSA) in Mathematical Epidemiology

The calculation of the probability of a minor outbreak is crucial in analyzing a stochastic epidemic model. For stochastic epidemic models with fixed delays, the linear chain trick is applied to transform the delayed models into a family of ODE models with increasing shape parameters. We then prove that the well-established results on the probability of a minor outbreak for continuous-time Markov chain (CTMC) epidemic models also hold for the stochastic epidemic models with fixed delays. All theoretical results are verified by numerical simulations implemented by the delay stochastic simulation algorithm (DSSA) in Python. It is shown that DSSA is able to generate exact realizations for underlying delayed models in the context of mathematical epidemiology, and therefore, provides insights into the effect of delays during the outbreak phases of epidemics.

A generalized stochastic SIR epidemic model with vaccination rules

Abstract In this paper, a generalized stochastic SIR epidemic model with vaccination rules is presented and the threshold behavior of the proposed epidemic model is investigated. Firstly, the stability of the equilibrium of the deterministic system is considered and the corresponding conditions are obtained. Secondly, the threshold of a stochastic SIR system for the extinction and the permanence in mean of epidemic disease are investigated. The results show that a larger stochastic disturbance can cause infections diseases to go to extinction. However, for a relatively small stochastic disturbance, the evolutionary dynamics of the epidemic diseases are overwhelmingly depend on the incidence function. This implies that the stochastic disturbance and the incidence function play an important role in diseases control. To test the theoretical results, a series of numerical simulations of these cases with respect to different noise disturbance coefficients are conducted.

Exit problem of stochastic SIR model with limited medical resource

Nonlinearity and randomness are both the essential attributes for the real world, and the case is the same for the models of infectious diseases, for which the deterministic models can not give a complete picture of the evolution. However, although there has been a lot of work on stochastic epidemic models, most of them focus mainly on qualitative properties, which makes us somewhat ignore the original meaning of the parameter value. In this paper we extend the classic susceptible-infectious-removed (SIR) epidemic model by adding a white noise excitation and then we utilize the large deviation theory to quantitatively study the long-term coexistence exit problem with epidemic. Finally, in order to extend the meaning of parameters in the corresponding deterministic system, we tentatively introduce two new thresholds which then prove rational.

Threshold dynamics of a stochastic SIQR epidemic model with imperfect quarantine

In this paper, we study a stochastic SIQR epidemic model with imperfect quarantine. We perform a detailed mathematical analysis of the dynamics of the stochastic model. The basic reproduction number R0s turns out to be a sharp threshold, that is, if R0s<1, then the disease-free equilibrium of the stochastic model is globally stable almost surely, whereas if R0s>1, the Markov process is positive recurrence which indicates that the disease will prevail. We also adopt the data of weekly infected cases for northern China from the first week of 2014 to the 26th week of 2014, to fit the model and estimate the parameters.

From individual-based epidemic models to McKendrick-von Foerster PDEs: a guide to modeling and inferring COVID-19 dynamics.

We present a unifying, tractable approach for studying the spread of viruses causing complex diseases requiring to be modeled using a large number of types (e.g., infective stage, clinical state, risk factor class). We show that recording each infected individual’s infection age, i.e., the time elapsed since infection, has three benefits. First, regardless of the number of types, the age distribution of the population can be described by means of a first-order, one-dimensional partial differential equation (PDE) known as the McKendrick-von Foerster equation. The frequency of type i is simply obtained by integrating the probability of being in state i at a given age against the age distribution. This representation induces a simple methodology based on the additional assumption of Poisson sampling to infer and forecast the epidemic. We illustrate this technique using French data from the COVID-19 epidemic. Second, our approach generalizes and simplifies standard compartmental models using high-dimensional systems of ordinary differential equations (ODEs) to account for disease complexity. We show that such models can always be rewritten in our framework, thus, providing a low-dimensional yet equivalent representation of these complex models. Third, beyond the simplicity of the approach, we show that our population model naturally appears as a universal scaling limit of a large class of fully stochastic individual-based epidemic models, where the initial condition of the PDE emerges as the limiting age structure of an exponentially growing population starting from a single individual.

Correlated stochastic epidemic model for the dynamics of SARS-CoV-2 with vaccination

In this paper, we propose a mathematical model to describe the influence of the SARS-CoV-2 virus with correlated sources of randomness and with vaccination. The total human population is divided into three groups susceptible, infected, and recovered. Each population group of the model is assumed to be subject to various types of randomness. We develop the correlated stochastic model by considering correlated Brownian motions for the population groups. As the environmental reservoir plays a weighty role in the transmission of the SARS-CoV-2 virus, our model encompasses a fourth stochastic differential equation representing the reservoir. Moreover, the vaccination of susceptible is also considered. Once the correlated stochastic model, the existence and uniqueness of a positive solution are discussed to show the problem’s feasibility. The SARS-CoV-2 extinction, as well as persistency, are also examined, and sufficient conditions resulted from our investigation. The theoretical results are supported through numerical/graphical findings.

STOCHASTIC OPTIMAL CONTROL ANALYSIS FOR THE COVID-19 EPIDEMIC MODEL UNDER REAL STATISTICS

The COVID-19 pandemic started, a global effort to develop vaccines and make them available to the public, has prompted a turning point in the history of vaccine development. In this study, we formulate a stochastic COVID-19 epidemic mathematical model with a vaccination effect. First, we present the model equilibria and basic reproduction number. To indicate that our stochastic model is well-posed, we prove the existence and uniqueness of a positive solution at the beginning. The sufficient conditions of the extinction and the existence of a stationary probability measure for the disease are established. For controlling the transmission of the disease by the application of external sources, the theory of stochastic optimality is established. The nonlinear least-squares procedure is utilized to parametrize the model from actual cases reported in Pakistan. The numerical simulations are carried out to demonstrate the analytical results.

Functional central limit theorems for epidemic models with varying infectivity

In this paper, we prove a functional central limit theorem (FCLT) for a stochastic epidemic model with varying infectivity and general infectious periods recently introduced in R. Forien et al. [Epidemic models with varying infectivity, SIAM J. Appl. Math. 81 (2021), pp. 1893–1930]. The infectivity process (total force of infection at each time) is composed of the independent infectivity random functions of each infectious individual, which starts at the time of infection. These infectivity random functions induce the infectious periods (as well as exposed, recovered or immune periods in full generality), whose probability distributions can be very general. The epidemic model includes the generalized non–Markovian SIR, SEIR, SIS, SIRS models with infection-age dependent infectivity. In the FCLTs for the generalized SIR and SEIR models, the limits of the diffusion-scaled fluctuations of the infectivity and susceptible processes are a unique solution to a two-dimensional Gaussian-driven stochastic Volterra integral equations, and then given these solutions, the limits for the infected (exposed/infectious) and recovered processes are Gaussian processes expressed in terms of the solutions to those stochastic Volterra integral equations. We also present the FCLTs for the generalized SIS and SIRS models.

Using Stochastic SIS Epidemic Model with Markovian Environment for Infectious Diseases

There are various types of stochastic epidemic models that can be formulated to deal with different types of diseases. In this project (SIS) epidemic model Susceptible - Infected- Susceptible will be considered relating to the Continuous Time Markov chains process. Each type of the epidemic models studies the disease according to its status. In particular, the SIS epidemic model regards infectious diseases. The Maple code is provided in this project, which is created to produce and predict the number of infected individuals and the disease prevalence at a fixed time.

Dynamical behavior of a stochastic SIQR epidemic model with Ornstein–Uhlenbeck process and standard incidence rate after dimensionality reduction

Due to the continuous interference of environmental white noise, the dynamical behaviors of a stochastic SIQR epidemic model with mean-reverting Ornstein–Uhlenbeck process and standard incidence are considered. Several conclusions can be verified after dimensionality reduction. We study the existence and uniqueness of positive solution by construct a nonnegative Lyapunov function at first. Then, a sufficient condition for extinction of the diseases is derived by constructing a suitable Lyapunov function. In addition, we also obtain the stationary distribution of the model by constructing a complex Lyapunov function. Particularly, we propose the exact local expression of the density function of the random model near the unique endemic equilibrium. Finally, the numerical simulations illustrate our above theoretical results.

Stochastic perturbation to 2-LTR dynamical model in HIV infected patients

In this paper, we proposed the stochastic perturbation to study the HIV viral dynamical model. The stochastic epidemic model of 2-LTR HIV infected patients is considered and we prove that the stochastic model admits a unique globally positive solution7 which is bounded and permanent. Then we analyze the necessary conditions for extinction and persistence of the disease by selecting the appropriate Lyapunov functions. The theoretical findings are confirmed by using suitable numerical experiments.

Threshold of Stochastic SIRS Epidemic Model from Infectious to Susceptible Class with Saturated Incidence Rate Using Spectral Method

Stochastic SIRS models play a key role in formulating and analyzing the transmission of infectious diseases. These models reflect the environmental changes of the diseases and their biological mechanisms. Therefore, it is very important to study the uniqueness and existence of the global positive solution to investigate the asymptotic properties of the model. In this article, we investigate the dynamics of the stochastic SIRS epidemic model with a saturated incidence rate. The effects of both deterministic and stochastic distribution from infectious to susceptible are analyzed. Our findings show that the occurrence of symmetry breaking as a function of the stochastic noise has a significant advantage over the deterministic one to prevent the spread of the infectious diseases. The larger stochastic noise will guarantee the control of epidemic diseases with symmetric Brownian motion. Periodic outbreaks and re-infection may occur due to the existence of feedback memory. It is shown that the endemic equilibrium is stable under some suitable initial conditions, taking advantage of the symmetry of the large amount of contact structure. A numerical method based on Legendre polynomials that converts the given stochastic SIRS model into a nonlinear algebraic system is used for the approximate solution. Finally, some numerical experiments are performed to verify the theoretical results and clearly show the sharpness of the obtained conditions and thresholds.

Analysis of Solution for the Stochastic Model Representing Water Scarcity in the Society

The main aim of this paper is to study the stochastic effect of water scarcity in the society through our model. It has been shown that there is a unique global positive solution to the proposed stochastic epidemic model with boundedness and permanence. We have selected some effective Lyapunov functions to provide sufficient conditions for investigating water scarcity persistence and extinction. The theoretical results of this work have been verified based on numerical experiments.

Stochastic Epidemic Model for COVID-19 Transmission under Intervention Strategies in China

In this paper, we discuss an EIQJR model with stochastic perturbation. First, a globally positive solution of the proposed model has been discussed. In addition, the global asymptotic stability and exponential mean-square stability of the disease-free equilibrium have been proven under suitable conditions for our model. This means that the disease will die over time. We investigate the asymptotic behavior around the endemic equilibrium of the deterministic model to show when the disease will prevail. Constructing a suitable Lyapunov functional method is crucial to our investigation. Parameter estimations and numerical simulations are performed to depict the transmission process of COVID-19 pandemic in China and to support analytical results.

A Numerical and Analytical Study of a Stochastic Epidemic SIR Model in the Light of White Noise

This study examines a novel SIR epidemic model that takes into account the impact of environmental white noise. According to the study, white noise has a significant impact on the disease. First, we establish the solution’s existence and uniqueness. Following that, we explain that the stochastic basic production R 0 is a threshold that determines the extinction or persistence of the disease. When noise levels are high, we acquire R 0 &lt; 1 , which causes the sickness to disappear. A sufficient condition for the existence of a stationary distribution is archived when the noise intensity is high, which suggests the infection is prevalent when R 0 &gt; 1 . Finally, numerical simulations are used to explain the key findings.

Analysis of a Stochastic SICR Epidemic Model Associated with the Lévy Jump

We propose and study a Susceptible-Infected-Confined-Recovered (SICR) epidemic model. For the proposed model, the driving forces include (for example) the Brownian motion processes and the jump Lévy noise. Usually, in the existing literature involving epidemiology models, the Lévy noise perturbations are ignored. However, in view of the presence of strong fluctuations in the SICR dynamics, it is worth including these perturbations in SICR epidemic models. Quite frequently, this results in several discontinuities in the processes under investigation. In our present study, we consider our SICR model after justifying its used form, namely, the component related to the Lévy noise. The existence and uniqueness of a global positive solution is established. Under some assumptions, we show the extinction and the persistence of the infection. In order to give some numerical simulations, we illustrate a new numerical method to validate our theoretical findings.