Stochastic Epidemic Model Research Articles

Stationary distribution and extinction of a stochastic generalized SEI epidemic model with Ornstein–Uhlenbeck process

In this paper, we propose a stochastic SEI epidemic model in which the transmission rates are general functions and satisfy the log-normal Ornstein–Uhlenbeck (OU) process. We first theoretically prove that there is a unique positive global solution of this stochastic model. By constructing several suitable Lyapunov functions, the sufficient condition R0s>1 is established for the existence of stationary distribution. The extinction of disease is also investigated and we find that the disease will die out at an exponential rate when R0E<1.

Threshold Analysis of a Stochastic SIRS Epidemic Model with Logistic Birth and Nonlinear Incidence

The paper mainly investigates a stochastic SIRS epidemic model with Logistic birth and nonlinear incidence. We obtain a new threshold value (R0m) through the Stratonovich stochastic differential equation, different from the usual basic reproduction number. If R0m&lt;1, the disease-free equilibrium of the illness is globally asymptotically stable in probability one. If R0m&gt;1, the disease is permanent in the mean with probability one and has an endemic stationary distribution. Numerical simulations are given to illustrate the theoretical results. Interestingly, we discovered that random fluctuations can suppress outbreaks and control the disease.

Dynamics and application of a generalized SIQR epidemic model with vaccination and treatment

In this paper, we propose and investigate the SIQR epidemic model with a generalized incidence rate function, a general treatment function and vaccination term. We firstly consider the existence and uniqueness of the global nonnegative solution to the deterministic model. Further, we show the locally asymptotic stability of the disease-free equilibrium and endemic equilibrium of the deterministic model, and obtain the basic reproduction number R0. Then we study the existence and uniqueness of the global positive solution to the stochastic model with any positive initial value. Meanwhile, 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. Finally, we make an empirical analysis by the COVID-19 data of Russia and Serbia. By the performance comparison of different models, it shows that the model with vaccination and treatment we proposed is better for the real situation, which is also verified by different estimation methods. Especially, that shows the recovery rate of the infected increases by 0.042 and the death rate of the recovered is 1.525 times that of normal human in Russia. Through statistical analysis, the short-term trend of epidemic transmission is predicted: under the condition of unchanged prevention and control policies, it may reach a stable endemic equilibrium state in Russia and the epidemic will eventually extinct in Serbia.

Stochastic Dynamics of a Virus Variant Epidemic Model with Double Inoculations

In this paper, we establish a random epidemic model with double vaccination and spontaneous variation of the virus. Firstly, we prove the global existence and uniqueness of positive solutions for a stochastic epidemic model. Secondly, we prove the threshold R0* can be used to control the stochastic dynamics of the model. If R0*&lt;0, the disease will be extinct with probability 1; whereas if R0*&gt;0, the disease can almost certainly continue to exist, and there is a unique stable distribution. Finally, we give some numerical examples to verify our theoretical results. Most of the existing studies prove the stochastic dynamics of the model by constructing Lyapunov functions. However, the construction of a Lyapunov function of higher-order models is extremely complex, so this method is not applicable to all models. In this paper, we use the definition method suitable for more models to prove the stationary distribution. Most of the stochastic infectious disease models studied now are second-order or third-order, and cannot accurately describe infectious diseases. In order to solve this kind of problem, this paper adopts a higher price five-order model.

Epidemic highs and lows: a stochastic diffusion model for active cases

We derive a stochastic epidemic model for the evolving density of infective individuals in a large population. Data shows main features of a typical epidemic consist of low periods interspersed with outbreaks of various intensities and duration. In our stochastic differential model, a novel reproductive term combines a factor expressing the recent notion of ‘attenuated Allee effect’ and a capacity factor is controlling the size of the process. Simulation of this model produces sample paths of the stochastic density of infectives, which behave much like long-time Covid-19 case data of recent years. Writing the process as a stochastic diffusion allows us to derive its stationary distribution, showing the relative time spent in low levels and in outbursts. Much of the behaviour of the density of infectives can be understood in terms of the interacting drift and diffusion coefficient processes, or, alternatively, in terms of the balance between noise level and the attenuation parameter of the Allee effect. Unexpected results involve the effect of increasing overall noise variance on the density of infectives, in particular on its level-crossing function.

Analysis of a stochastic SVIR model with time‐delayed stages of vaccination and Lévy jumps

The focal point of this paper is to further enhance the existing stochastic epidemic models by incorporating several new disease characteristics, such as the validation time of the vaccination procedure, the stages of vaccine required to gain a long‐period immunity together with the time separating each stage, the deaths linked to the vaccine, and finally, the sudden environmental noise which is exhibited by sociocultural changes, such as antivaccination movements. To incorporate all the aforementioned characteristics, we extend the standard Susceptible‐Vaccinated‐Infected‐Recovered (SVIR) epidemic model to a new mathematical model, which is governed by a system of coupled stochastic delay differential equations, in which the disease transmission rates are driven by Gaussian noise and Lévy‐type jump stochastic process. First, under suitable conditions on the jump intensities, we address the mathematical well‐posedness and biological feasibility of the model, by virtue of the Lyapunov method and the stopping‐time technique. Then, by choosing an adequate positively invariant set for the considered model, we establish sufficient conditions guaranteeing the disease extinction and persistence. Lastly, to support the theoretical results, we provide the outcome of several numerical simulations which, together with our conducted analysis, indicate that the spread of the disease can be majorly altered by all the new considered characteristics.

Fitting stochastic epidemic models to gene genealogies using linear noise approximation.

Phylodynamics is a set of population genetics tools that aim at reconstructing demographic history of a population based on molecular sequences of individuals sampled from the population of interest. One important task in phylodynamics is to estimate changes in (effective) population size. When applied to infectious disease sequences such estimation of population size trajectories can provide information about changes in the number of infections. To model changes in the number of infected individuals, current phylodynamic methods use non-parametric approaches (e.g., Bayesian curve-fitting based on change-point models or Gaussian process priors), parametric approaches (e.g., based on differential equations), and stochastic modeling in conjunction with likelihood-free Bayesian methods. The first class of methods yields results that are hard to interpret epidemiologically. The second class of methods provides estimates of important epidemiological parameters, such as infection and removal/recovery rates, but ignores variation in the dynamics of infectious disease spread. The third class of methods is the most advantageous statistically, but relies on computationally intensive particle filtering techniques that limits its applications. We propose a Bayesian model that combines phylodynamic inference and stochastic epidemic models, and achieves computational tractability by using a linear noise approximation (LNA) - a technique that allows us to approximate probability densities of stochastic epidemic model trajectories. LNA opens the door for using modern Markov chain Monte Carlo tools to approximate the joint posterior distribution of the disease transmission parameters and of high dimensional vectors describing unobserved changes in the stochastic epidemic model compartment sizes (e.g., numbers of infectious and susceptible individuals). In a simulation study, we show that our method can successfully recover parameters of stochastic epidemic models. We apply our estimation technique to Ebola genealogies estimated using viral genetic data from the 2014 epidemic in Sierra Leone and Liberia.

A generalized stochastic SIRS epidemic model incorporating mean-reverting Ornstein–Uhlenbeck process

The aim of this work is to study a new stochastic SIRS epidemic model that includes the mean-reverting Ornstein–Uhlenbeck process and a general incidence rate. First, we prove the global existence and positivity of the solution by using Lyapunov functions. Second, we analytically make out the stochastic epidemic threshold T̃0S which pilots the extinction and persistence in mean of the disease. We have proven that the disease extinguishes when T̃0S<1. Otherwise, if T̃0S>1, then disease is persistent in mean. For the critical case T̃0S=1, we have shown that the disease dies out by using an approach involving some appropriate stopping times. Finally, we present a series of numerical simulations to confirm the feasibility and correctness of the theoretical analysis results.

Asymptotic behavior of an SIQR epidemic model driven by Lévy jumps on scale-free networks

To investigate the effect of information transmission, Lévy jumps and contact heterogeneity of individuals on the asymptotic behavior of epidemic, a stochastic SIQR epidemic model with non-monotone incidence rate and Lévy jumps on scale-free networks is constructed. At first, the global dynamics of the deterministic model is studied by constructing appropriate Lyapunov functions. Then the stochastic model is made in accordance with the ecological significance, the existence and uniqueness of the global positive solution of the stochastic SIQR model is manifested. Next, by constructing suitable stochastic Lyapunov functions and applying Itô formula with jump, the asymptotic behavior of solutions of stochastic model around equilibrium of the corresponding deterministic model is checked. At last, the correctness of the analytical results is verified by numerical simulations.

Threshold Dynamics and Probability Density Function of a Stochastic Avian Influenza Epidemic Model with Nonlinear Incidence Rate and Psychological Effect

Threshold Dynamics and Probability Density Function of a Stochastic Avian Influenza Epidemic Model with Nonlinear Incidence Rate and Psychological Effect

Vaccination effect on a stochastic epidemic model with healing and relapse

In this work, we consider a stochastic epidemic model with vaccination, healing and relapse. We prove the existence and the uniqueness of the positive solution. We establish sufficient conditions for the extinction and the persistence in mean of the stochastic system. Moreover, we also establish sufficient conditions for the existence of ergodic stationary distribution to the model, which reveals that the infectious disease will persist. The graphical illustrations of the approximate solutions of the stochastic epidemic model have been performed.

A novel mathematical analysis and threshold reinforcement of a stochastic dengue epidemic model with Lévy jumps

The aggressive re-emergence of dengue fever during the last decade made it a principal topic of several research fields, especially biomathematics. In this vein, mathematical modeling is commonly viewed as a great option to get an idea of this disease’s prevalence behavior. This work presents and analyzes a generalized stochastic dengue model that incorporates slight and huge environmental perturbations. In rigorous terms, our proposed model takes the form of a stochastic differential equations system that includes white noises and Lévy jumps. We first demonstrate its well-posedness then, and based on many novel analytical techniques, we provide some ameliorated necessary conditions for the extinction and the persistence in the mean. The theoretical findings show that the dynamics of our disturbed dengue model are mainly determined by the parameters related to the white noises and Lévy jumps intensities. In the end, certain numerical illustrative examples are depicted to confirm our findings and to highlight the effect of the adopted mathematical techniques on the results.

Euler-Maruyama and Kloeden-Platen-Schurz computing paradigm for stochastic vector-borne plant epidemic model

The dynamics of viral infection within-plant hosts are of critical importance for characterizing the prevalence and impact of plant diseases. However, few mathematical modeling efforts have been made to characterize viral dynamics within plants. In this study, the dynamics of the vector-borne plant epidemic (VBPE) model are modeled with two nonlinear mathematical models, the deterministic vector-borne plant epidemic (DVBPE) model and the stochastic vector-borne plant epidemic (SVBPE) model to portray and forecast the virus dynamical behavior. The VBPE model segregated the population into three classes: susceptible plants (S), infectious plants (I), and infectious vectors (Y). For classes S, I, and Y, the approximate solution is established by creating a sufficient number of scenarios by varying the ratio of infection, infected vector biting rate, infection rates of plant model, host’s capacity of infection, disease-induced death rate of infectious hosts, insect vectors’ natural mortality rate, the stochastic term for susceptible plants, the stochastic term for infected plants and stochastic term for infected vectors The Adams method is utilized to determine the approximate solution for the DVBPE model, while the Euler-Maruyama and Kloeden-Platen-Schurz methods are used to evaluate the SVBPE model. Finally, a comparison between the DVBPE and SVBPE models is presented.

Mathematical analysis of stochastic epidemic model of MERS-corona & application of ergodic theory

The “Middle East Respiratory” (MERS-Cov) is among the world’s dangerous diseases that still exist. Presently it is a threat to Arab countries, but it is a horrible prediction that it may propagate like COVID-19. In this article, a stochastic version of the epidemic model, MERS-Cov, is presented. Initially, a mathematical form is given to the dynamics of the disease while incorporating some unpredictable factors. The study of the underlying model shows the existence of positive global solution. Formulating appropriate Lyapunov functionals, the paper will also explore parametric conditions which will lead to the extinction of the disease from a community. Moreover, to reveal that the infection will persist, ergodic stationary distribution will be carried out. It will also be shown that a threshold quantity exists, which will determine some essential parameters for exploring other dynamical aspects of the main model. With the addition of some examples, the underlying stochastic model of MERS-Cov will be studied graphically for more illustration.

Forecast analysis and sliding mode control on a stochastic epidemic model with alertness and vaccination

In this paper, a stochastic SEIR epidemic model is studied with alertness and vaccination. The goal is to stabilize the infectious disease system quickly. The dynamic behavior of the model is analyzed and an integral sliding mode controller with distributed compensation is designed. By using Lyapunov function method, the sufficient conditions for the existence and uniqueness of global positive solutions and the existence of ergodic stationary distributions are obtained. The stochastic center manifold and stochastic average method are used to simplify the system into a one-dimensional Markov diffusion process. The stochastic stability and Hopf bifurcation are analyzed using singular boundary theory. An integral sliding mode controller with non-parallel distributed compensation is designed by linear matrix inequality (LMI) method, which realizes the stability of system and prevents the outbreak of epidemic disease. The correction of theoretical analysis and the effectiveness of controller are validated using numerical simulation performed in MATLAB/Simulink.

Global dynamics and density function in a class of stochastic SVI epidemic models with Lévy jumps and nonlinear incidence

&lt;abstract&gt;&lt;p&gt;The paper studies the global dynamics and probability density function for a class of stochastic SVI epidemic models with white noise, Lévy jumps and nonlinear incidence. The stability of disease-free and endemic equilibria for the corresponding deterministic model is first obtained. The threshold criteria on the stochastic extinction, persistence and stationary distribution are established. That is, the disease is extinct with probability one if the threshold value $ R_{0}^{s} &amp;lt; 1 $, and the disease is persistent in the mean and any positive solution is ergodic and has a unique stationary distribution if $ R_{0}^{s} &amp;gt; 1 $. Furthermore, the approximate expression of the log-normal probability density function around the quasi-endemic equilibrium of the stochastic model is calculated. A new technique for the calculation of the probability density function is proposed. Lastly, the numerical examples and simulations are presented to verify the main results.&lt;/p&gt;&lt;/abstract&gt;

Threshold dynamics of stochastic cholera epidemic model with direct transmission

&lt;abstract&gt;&lt;p&gt;This paper extends the cholera human-to-human direct transmission model from a deterministic to a stochastic framework. This is expressed as mixed system of stochastic and deterministic differential equations. A Lyapunov function is created to investigate the global stability of the stochastic cholera epidemic, which shows the existence of global positivity of the solution using the theory of stopping time. We then find the threshold quantity of the extended stochastic cholera epidemic model. We derive a parametric condition $ \widetilde{R}_0 $, and for additive white noise, we establish sufficient conditions for the extinction and the persistence of the cholera infection. Finally, for a suitable choice of the parameter of the system for $ \widetilde{R}_0 $, we perform numerical simulations for both scenarios of extinction and persistence of the dynamic of the cholera infection.&lt;/p&gt;&lt;/abstract&gt;

Threshold dynamics of a stochastic general SIRS epidemic model with migration.

In this study, a stochastic SIRS epidemic model that features constant immigration and general incidence rate is investigated. Our findings show that the dynamical behaviors of the stochastic system can be predicted using the stochastic threshold $ R_0^S $. If $ R_0^S < 1 $, the disease will become extinct with certainty, given additional conditions. Conversely, if $ R_0^S > 1 $, the disease has the potential to persist. Moreover, the necessary conditions for the existence of the stationary distribution of positive solution in the event of disease persistence is determined. Our theoretical findings are validated through numerical simulations.

A mathematical study of the COVID-19 propagation through a stochastic epidemic model

The COVID-19 is a major danger that threatens the whole world. In this context, mathematical modeling is a very powerful tool for knowing more about how such a disease is transmitted within a host population of humans. In this regard, we propose in the current study a stochastic epidemic model that describes the COVID-19 dynamics under the application of quarantine and coverage media strategies, and we give a rigorous mathematical analysis of this model to obtain an overview of COVID-19 dissemination behavior.

Dynamics of a stochastic epidemic model with quarantine and non-monotone incidence

&lt;abstract&gt;&lt;p&gt;In this paper, a stochastic SIQR epidemic model with non-monotone incidence is investigated. First of all, we consider the disease-free equilibrium of the deterministic model is globally asymptotically stable by using the Lyapunov method. Secondly, the existence and uniqueness of positive solution to the stochastic model is obtained. Then, the sufficient condition for extinction of the stochastic model is established. Furthermore, a unique stationary distribution to stochastic model will exist by constructing proper Lyapunov function. Finally, numerical examples are carried out to illustrate the theoretical results, with the help of numerical simulations, we can see that the higher intensities of the white noise or the bigger of the quarantine rate can accelerate the extinction of the disease. This theoretically explains the significance of quarantine strength (or isolation measures) when an epidemic erupts.&lt;/p&gt;&lt;/abstract&gt;