Stochastic SIR Model Research Articles

The epidemiological footprint of contact structures in models with two levels of mixing.

Models with several levels of mixing (households, workplaces), as well as various corresponding formulations for , have been proposed in the literature. However, little attention has been paid to the impact of the distribution of the population size within social structures, effect that can help plan effective interventions. We focus on the influence on the model outcomes of teleworking strategies, consisting in reshaping the distribution of workplace sizes. We consider a stochastic SIR model with two levels of mixing, accounting for a uniformly mixing general population, each individual belonging also to a household and a workplace. The variance of the workplace size distribution appears to be a good proxy for the impact of this distribution on key outcomes of the epidemic, such as epidemic size and peak. In particular, our findings suggest that strategies where the proportion of individuals teleworking depends sublinearly on the size of the workplace outperform the strategy with linear dependence. Besides, one drawback of the model with multiple levels of mixing is its complexity, raising interest in a reduced model. We propose a homogeneously mixing SIR ODE-based model, whose infection rate is chosen as to observe the growth rate of the initial model. This reduced model yields a generally satisfying approximation of the epidemic. These results, robust to various changes in model structure, are very promising from the perspective of implementing effective strategies based on social distancing of specific contacts. Furthermore, they contribute to the effort of building relevant approximations of individual based models at intermediate scales.

Parameter estimation for stochastic SIR model

In this article, we study the parameter estimation problem of the continuous-time continuous-state stochastic SIR model in the context of discrete-time observations. Specifically, we propose estimators of the efficient contact rate and recovery rate based on weighted conditional least squares estimators (WCLSEs) and establish the relevant asymptotic distribution of the corresponding estimators under appropriate moment conditions.

A unified stochastic SIR model driven by Lévy noise with time-dependency

We propose a unified stochastic SIR model driven by Lévy noise. The model is structural enough to allow for time-dependency, nonlinearity, discontinuity, demography, and environmental disturbances. We present concise results on the existence and uniqueness of positive global solutions and investigate the extinction and persistence of the novel model. Examples and simulations are provided to illustrate the main results.

Nonlinear Dynamics of a General Stochastic SIR Model with Behavioral and Physical Changes: Analysis and Application to Zoonotic Tuberculosis

This paper presents a comprehensive nonlinear analysis of an innovative stochastic epidemic model that accounts for both behavioral changes and physical discontinuities. Our research begins with the formulation of a perturbed model, integrating two general incidence functions and incorporating a Lévy measure to account for independent jump components. We start by confirming the well-posed nature of the model, ensuring its mathematical soundness and feasibility for further analysis. Following this, we establish a global threshold criterion that serves to distinguish between the eradication and the persistence of an epidemic. This threshold is crucial for understanding the long-term behavior of a disease within a population. To rigorously validate the accuracy of this threshold, we conducted extensive numerical simulations using estimated data on Zoonotic Tuberculosis in Morocco. These simulations provide practical insights and reinforce the theoretical findings of our study. A notable aspect of our approach is its significant advancement over previous works in the literature. Our model not only offers a more comprehensive framework but also identifies optimal conditions under which an epidemic can be controlled or eradicated.

Critical Behavior of the Stochastic SIR Model on Random Bond-Diluted Lattices

Critical Behavior of the Stochastic SIR Model on Random Bond-Diluted Lattices

Disease X epidemic control using a stochastic model and a deterministic approximation: Performance comparison with and without parameter uncertainties

Background: The spread of infectious diseases can be modeled using deterministic or stochastic models. A deterministic approximation of a stochastic model can be appropriate under some conditions, but is unable to capture the discrete nature of populations. We look into the choice of a model from the perspective of decision making.Method: We consider an emerging disease (Disease X) in a closed population modeled by a stochastic SIR model or its deterministic approximation. The objective of the decision maker is to minimize the cumulative number of symptomatic infected-days over the course of the epidemic by picking a vaccination policy. We consider four decision making scenarios: based on the stochastic model or the deterministic model, and with or without parameter uncertainty. We also consider different sample sizes for uncertain parameter draws and stochastic model runs. We estimate the average performance of decision making in each scenario and for each sample size.Results: The model used for decision making has an influence on the picked policies. The best achievable performance is obtained with the stochastic model, knowing parameter values, and for a large sample size. For small sample sizes, the deterministic model can outperform the stochastic model due to stochastic effects. Resolving uncertainties may bring more benefit than switching to the stochastic model in our example.Conclusion: This article illustrates the interplay between the choice of a type of model, parameter uncertainties, and sample sizes. It points to issues to be considered when optimizing a stochastic model.

Differentiated uniformization: a new method for inferring Markov chains on combinatorial state spaces including stochastic epidemic models

AbstractWe consider continuous-time Markov chains that describe the stochastic evolution of a dynamical system by a transition-rate matrix Q which depends on a parameter $$\theta $$ θ . Computing the probability distribution over states at time t requires the matrix exponential $$\exp \,\left( tQ\right) \,$$ exp t Q , and inferring $$\theta $$ θ from data requires its derivative $$\partial \exp \,\left( tQ\right) \,/\partial \theta $$ ∂ exp t Q / ∂ θ . Both are challenging to compute when the state space and hence the size of Q is huge. This can happen when the state space consists of all combinations of the values of several interacting discrete variables. Often it is even impossible to store Q. However, when Q can be written as a sum of tensor products, computing $$\exp \,\left( tQ\right) \,$$ exp t Q becomes feasible by the uniformization method, which does not require explicit storage of Q. Here we provide an analogous algorithm for computing $$\partial \exp \,\left( tQ\right) \,/\partial \theta $$ ∂ exp t Q / ∂ θ , the differentiated uniformization method. We demonstrate our algorithm for the stochastic SIR model of epidemic spread, for which we show that Q can be written as a sum of tensor products. We estimate monthly infection and recovery rates during the first wave of the COVID-19 pandemic in Austria and quantify their uncertainty in a full Bayesian analysis. Implementation and data are available at https://github.com/spang-lab/TenSIR.

Threshold dynamics of a switching diffusion SIR model with logistic growth and healthcare resources.

In this article, we have constructed a stochastic SIR model with healthcare resources and logistic growth, aiming to explore the effect of random environment and healthcare resources on disease transmission dynamics. We have showed that under mild extra conditions, there exists a critical parameter, i.e., the basic reproduction number $ R_0^s $, which completely determines the dynamics of disease: when $ R_0^s < 1 $, the disease is eradicated; while when $ R_0^s > 1 $, the disease is persistent. To validate our theoretical findings, we conducted some numerical simulations using actual parameter values of COVID-19. Both our theoretical and simulation results indicated that (1) the white noise can significantly affect the dynamics of a disease, and importantly, it can shift the stability of the disease-free equilibrium; (2) infectious disease resurgence may be caused by random switching of the environment; and (3) it is vital to maintain adequate healthcare resources to control the spread of disease.

Threshold behavior and exponential ergodicity of an sir epidemic model: the impact of random jamming and hospital capacity.

This article uses hospital capacity to determine the treatment rate for an infectious disease. To examine the impact of random jamming and hospital capacity on the spread of the disease, we propose a stochastic SIR model with nonlinear treatment rate and degenerate diffusion. Our findings demonstrate that the disease's persistence or eradication depends on the basic reproduction number [Formula: see text]. If [Formula: see text], the disease is eradicated with a probability of 1, while [Formula: see text] results in the disease being almost surely strongly stochastically permanent. We also demonstrate that if [Formula: see text], the Markov process has a unique stationary distribution and is exponentially ergodic. Additionally, we identify a critical capacity which determines the minimum hospital capacity required.

Risky driving behavior propagation: A novel stochastic SIR model and two-stage risk quantification method

Risky driving behavior propagation: A novel stochastic SIR model and two-stage risk quantification method

Randomness accelerates the dynamic clearing process of the COVID-19 outbreaks in China

During the implementation of strong non-pharmaceutical interventions (NPIs), more than one hundred COVID-19 outbreaks induced by different strains in China were dynamically cleared in about 40 days, which presented the characteristics of small scale clustered outbreaks with low peak levels. To address how did randomness affect the dynamic clearing process, we derived an iterative stochastic difference equation for the number of newly reported cases based on the classical stochastic SIR model and calculate the stochastic control reproduction number (SCRN). Further, by employing the Bayesian technique, the change points of SCRNs have been estimated, which is an important prerequisite for determining the lengths of the exponential growth and decline phases. To reveal the influence of randomness on the dynamic zeroing process, we calculated the explicit expression of the mean first passage time (MFPT) during the decreasing phase using the relevant theory of first passage time (FPT), and the main results indicate that random noise can accelerate the dynamic zeroing process. This demonstrates that powerful NPI measures can rapidly reduce the number of infected people during the exponential decline phase, and enhanced randomness is conducive to dynamic zeroing, i.e. the greater the random noise, the shorter the average clearing time is. To confirm this, we chose 26 COVID-19 outbreaks in various provinces in China and fitted the data by estimating the parameters and change points. We then calculated the MFPTs, which were consistent with the actual duration of dynamic zeroing interventions.

Computational aspects of an epidemic model involving stochastic partial differential equations

This paper deals with the study of the reaction–diffusion epidemic model perturbed with time noise. It has various applications such as disease in population models of humans, wildlife, and many others. The stochastic SIR model is numerically investigated with the proposed stochastic backward Euler scheme and proposed stochastic implicit finite difference (IFD) scheme. The stability of the proposed methods is shown with Von Neumann criteria and both schemes are unconditionally stable. Both schemes are consistent with systems of the equations in the mean square sense. The numerical solution obtained by the proposed stochastic backward Euler scheme and solutions converges towards an equilibrium but it has negative and divergent behavior for some values. The numerical solution gained by the proposed IFD scheme preserves the positivity and also solutions converge towards endemic and disease-free equilibrium. We have used two problems to check our findings. The graphical behavior of the stochastic SIR model is much adjacent to the classical SIR epidemic model when noise strength approaches zero. The three-dimensional plots of the susceptible and infected individuals are drawn for two cases of endemic equilibrium and disease-free equilibriums. The results show the efficacy of the proposed stochastic IFD scheme.

SIR epidemics driven by Feller processes

AbstractWe consider a stochastic SIR (susceptible $\rightarrow$ infective $\rightarrow$ removed) model in which the infectious periods are modulated by a collection of independent and identically distributed Feller processes. Each infected individual is associated with one of these processes, the trajectories of which determine the duration of his infectious period, his contamination rate, and his type of removal (e.g. death or immunization). We use a martingale approach to derive the distribution of the final epidemic size and severity for this model and provide some general examples. Next, we focus on a single infected individual facing a given number of susceptibles, and we determine the distribution of his outcome (number of contaminations, severity, type of removal). Using a discrete-time formulation of the model, we show that this distribution also provides us with an alternative, more stable method to compute the final epidemic outcome distribution.

Hybrid stochastic epidemic SIR models with hidden states

This paper focuses on realistic hybrid SIR models that take into account stochasticity. The proposed systems are applicable to most incidence rates that are used in the literature including the bilinear incidence rate, the Beddington–DeAngelis incidence rate, and a Holling type II functional response. Given that many diseases can lead to asymptomatic infections, this paper looks at a system of stochastic differential equations that also includes a class of hidden state individuals, for which the infection status is unknown. Assuming that the direct observation of the percentage of hidden state individuals being infected, α(t), is not given and only a noise-corrupted observation process is available. Using nonlinear filtering techniques in conjunction with an invasion type analysis, this paper shows that the long-term behavior of the disease is governed by a threshold λ∈R that depends on the model parameters. It turns out that if λ<0 the number I(t) of infected individuals converges to zero exponentially fast (extinction). However, if λ>0, the infection is endemic and the system is persistent. We showcase our theorems by applying them in some illuminating examples.

A dynamical study on stochastic reaction diffusion epidemic model with nonlinear incidence rate.

The current study deals with the stochastic reaction-diffusion epidemic model numerically with two proposed schemes. Such models have many applications in the disease dynamics of wildlife, human life, and others. During the last decade, it is observed that the epidemic models cannot predict the accurate behavior of infectious diseases. The empirical data gained about the spread of the disease shows non-deterministic behavior. It is a strong challenge for researchers to consider stochastic epidemic models. The effect of the stochastic process is analyzed. So, the SIR epidemic model is considered under the influence of the stochastic process. The time noise term is taken as the stochastic source. The coefficient of the stochastic term is a Borel function, and it is used to control the random behavior in the solutions. The proposed stochastic backward Euler scheme and the proposed stochastic implicit finite difference scheme (IFDS) are used for the numerical solution of the underlying model. Both schemes are consistent in the mean square sense. The stability of the schemes is proven with Von-Neumann criteria and schemes are unconditionally stable. The proposed stochastic backward Euler scheme converges toward a disease-free equilibrium and does not converge toward an endemic equilibrium but also possesses negative behavior. The proposed stochastic IFD scheme converges toward disease-free equilibrium and endemic equilibrium. This scheme also preserves positivity. The graphical behavior of the stochastic SIR model is much similar to the classical SIR epidemic model when noise strength approaches zero. The three-dimensional plots of the susceptible and infected individuals are drawn for two cases of endemic equilibrium and disease-free equilibriums. The efficacy of the proposed scheme is shown in the graphical behavior of the test problem for the various values of the parameters.

Variants of the Stochastic Sir Models and Vaccination Strategies.

Several options of the stochastic SIR epidemics model with limited treatment are proposed. For these methods, the efficiency of different vaccination strategies is demonstrated, and a method for obtaining the optimal vaccination strategy minimizing the cost functional is proposed.

Numerical analysis of the Linearly implicit Euler method with truncated Wiener process for the stochastic SIR model

The paper deals with the numerical positivity, convergence and dynamical behaviors (including extinction and persistence) for stochastic SIR model. For the real significance of the numerical analysis on stochastic SIR model, a linearly implicit Euler method with truncated Wiener process is introduced. The numerical positivity is obtained by the truncated Wiener process, which is the basis for the investigation of convergence and dynamical behavior. The numerical dynamical behavior is obtained by an exponential presentation for the nonlinear stochastic stability function and the large number theorem for martingale, which reproduces the existing theoretical results of exact solution. Finally, numerical examples are given to validate our numerical results for stochastic SIR model.

Dynamic behavior of a stochastic SIR model with nonlinear incidence and recovery rates

&lt;abstract&gt;&lt;p&gt;The spread of infectious diseases are inevitably affected by natural and social factors, and their evolution presents oscillations and other uncertainties. Therefore, it is of practical significance to consider stochastic noise interference in the studies of infectious disease models. In this paper, a stochastic SIR model with nonlinear incidence and recovery rate is studied. First, a unique global positive solution for any initial value of the system is proved. Second, we provide the sufficient conditions for disease extinction or persistence, and the influence of threshold $ \tilde{R_{0}} $ of the stochastic SIR model on disease state transition is analyzed. Additionally, we prove that the system has a stationary distribution under some given parameter conditions by building an appropriate stochastic Lyapunov function as well as using the equivalent condition of the Hasminskii theorem. Finally, the correctness of these theoretical results are validated by numerical simulations.&lt;/p&gt;&lt;/abstract&gt;

Application of the Stochastic SIR Model for Simulation of the Epidemic Process

Application of the Stochastic SIR Model for Simulation of the Epidemic Process

Stochastic SIR model predicts the evolution of COVID-19 epidemics from public health and wastewater data in small and medium-sized municipalities: A one year study

The level of unpredictability of the COVID-19 pandemics poses a challenge to effectively model its dynamic evolution. In this study we incorporate the inherent stochasticity of the SARS-CoV-2 virus spread by reinterpreting the classical compartmental models of infectious diseases (SIR type) as chemical reaction systems modeled via the Chemical Master Equation and solved by Monte Carlo Methods. Our model predicts the evolution of the pandemics at the level of municipalities, incorporating for the first time (i) a variable infection rate to capture the effect of mitigation policies on the dynamic evolution of the pandemics (ii) SIR-with-jumps taking into account the possibility of multiple infections from a single infected person and (iii) data of viral load quantified by RT-qPCR from samples taken from Wastewater Treatment Plants. The model has been successfully employed for the prediction of the COVID-19 pandemics evolution in small and medium size municipalities of Galicia (Northwest of Spain).