scholarly journals State space collapse for critical multistage epidemics

2015 ◽  
Vol 47 (03) ◽  
pp. 715-740
Author(s):  
Florian Simatos
Keyword(s):  
State Space ◽  
Epidemic Model ◽  
Infected Individual ◽  
Sir Model ◽  
Critical Regime ◽  
Susceptible Individual ◽  
Diffusion Approximations ◽  
Unusual Form

We study a multistage epidemic model which generalizes the SIR model and where infected individuals go through K ≥ 1 stages of the epidemic before being removed. An infected individual in stage k ∈ {1, …, K} may infect a susceptible individual, who directly goes to stage k of the epidemic; or it may go to the next stage k + 1 of the epidemic. For this model, we identify the critical regime in which we establish diffusion approximations. Surprisingly, the limiting diffusion exhibits an unusual form of state space collapse which we analyze in detail.

scholarly journals State space collapse for critical multistage epidemics

2015 ◽  
Vol 47 (3) ◽  
pp. 715-740
Author(s):  
Florian Simatos
Keyword(s):  
State Space ◽  
Epidemic Model ◽  
Infected Individual ◽  
Sir Model ◽  
Critical Regime ◽  
Susceptible Individual ◽  
Diffusion Approximations ◽  
Unusual Form

We study a multistage epidemic model which generalizes the SIR model and where infected individuals go through K ≥ 1 stages of the epidemic before being removed. An infected individual in stage k ∈ {1, …, K} may infect a susceptible individual, who directly goes to stage k of the epidemic; or it may go to the next stage k + 1 of the epidemic. For this model, we identify the critical regime in which we establish diffusion approximations. Surprisingly, the limiting diffusion exhibits an unusual form of state space collapse which we analyze in detail.

scholarly journals Fractional-Order SIR Epidemic Model for Transmission Prediction of COVID-19 Disease

2020 ◽  
Vol 10 (23) ◽  
pp. 8316
Author(s):  
Kamil Kozioł ◽  
Rafał Stanisławski ◽  
Grzegorz Bialic
Keyword(s):  
Fractional Order ◽  
Epidemic Model ◽  
Dynamic Properties ◽  
Sir Model ◽  
Domain Model ◽  
Sir Epidemic Model ◽  
Step Method ◽  
Order Difference ◽  
Sir Epidemic ◽  
The Time Domain

In this paper, the fractional-order generalization of the susceptible-infected-recovered (SIR) epidemic model for predicting the spread of the COVID-19 disease is presented. The time-domain model implementation is based on the fixed-step method using the nabla fractional-order difference defined by Grünwald-Letnikov formula. We study the influence of fractional order values on the dynamic properties of the proposed fractional-order SIR model. In modeling the COVID-19 transmission, the model’s parameters are estimated while using the genetic algorithm. The model prediction results for the spread of COVID-19 in Italy and Spain confirm the usefulness of the introduced methodology.

scholarly journals A Survey on Two Viruses Extensions of Epidemic Model with Continuous and Impulse Control

2021 ◽  
Vol 14 ◽  
pp. 127-154
Author(s):  
Elena Gubar ◽  
◽  
Vladislav Taynitskiy ◽  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Network Structure ◽  
Epidemic Model ◽  
Impulse Control ◽  
Sir Model ◽  
Initial Population ◽  
Protection Measures ◽  
Virus Spreading ◽  
Theoretical Results

The current study represents a survey on several modifications of compartment epidemic models with continuous and impulse control policies. The main contribution of the survey is the modification of the classical Susceptible Infected Recovered (SIR) model with the assumption that two types of viruses are circulating in the population at the same time. Moreover, we also take into consideration the network structure of the initial population in two-virus SIIR models and estimate the e ectiveness of protection measures over complex networks. In each model, the optimal control problem has been formalized to minimize the costs of the virus spreading and find optimal continuous and impulse antivirus controllers. All theoretical results are corroborated by a large number of numerical simulations.

scholarly journals An Age and Space Structured SIR Model Describing the Covid-19 Pandemic

2020 ◽  
Author(s):  
Rinaldo M Colombo ◽  
Mauro Garavello ◽  
Francesca Marcellini ◽  
Elena Rossi
Keyword(s):  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Sir Model ◽  
Qualitative Properties ◽  
Numerical Integrations ◽  
Key Features ◽  
Virus Spreading ◽  
The Basic Reproduction Number

We present an epidemic model capable of describing key features of the present Covid-19 pandemic. While capturing several qualitative properties of the virus spreading, it allows to compute the basic reproduction number, the number of deaths due to the virus and various other statistics. Numerical integrations are used to illustrate the relevance of quarantine and the role of care houses.

Gaussian approximation of some closed stochastic epidemic models

1977 ◽  
Vol 14 (02) ◽  
pp. 221-231
Author(s):  
Frank J. S. Wang
Keyword(s):  
Epidemic Model ◽  
Limit Theorems ◽  
Gaussian Approximation ◽  
Infected Individual ◽  
Small Time ◽  
Entire Population ◽  
Time Interval ◽  
Stochastic Epidemic Models ◽  
Small Time Interval ◽  
Stochastic Epidemic

A generalization of Bailey's general epidemic model is considered. In this generalized model, it is assumed that the probability of any particular susceptible becoming infected during the small time interval (t, t + Δt) is α(X(t))Δt + o(Δt), for some function a, where X(t) is the proportion of infected individuals in the entire population, the probability that an infected individual is infected for at least a length of time t is F(t), and recovered individuals are permanently immune from further attack. In this paper, central limit theorems are obtained for the proportion of infected individuals and the proportion of susceptibles in the entire population.

Corrected Diffusion Approximations for Ruin Probabilities in a Markov Random Walk

1997 ◽  
Vol 29 (03) ◽  
pp. 695-712 ◽  
Author(s):  
C. D. Fuh
Keyword(s):  
Random Walk ◽  
State Space ◽  
Stopping Times ◽  
Ruin Probabilities ◽  
Diffusion Approximations ◽  
Markov Random ◽  
Markov Random Walk ◽  
Finite State ◽  
Correction Terms ◽  
Finite State Space

Let (X, S) = {(Xn , Sn ); n ≧0} be a Markov random walk with finite state space. For a ≦ 0 < b define the stopping times τ= inf {n:Sn > b} and T= inf{n:Sn ∉(a, b)}. The diffusion approximations of a one-barrier probability P {τ < ∝ | X o = i}, and a two-barrier probability P{ST ≧b | X o = i} with correction terms are derived. Furthermore, to approximate the above ruin probabilities, the limiting distributions of overshoot for a driftless Markov random walk are involved.

DYNAMIC ANALYSIS OF AN SIR MODEL WITH STOCHASTIC PERTURBATIONS

2013 ◽  
Vol 06 (06) ◽  
pp. 1350041
Author(s):  
ZHENJIE LIU ◽  
JINLIANG WANG ◽  
YALAN XU ◽  
GUOQIANG LI
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Sir Model ◽  
Sir Epidemic Model ◽  
Stochastic Perturbations ◽  
Sir Epidemic ◽  
Stochastic Epidemic ◽  
Differential Infectivity ◽  
Global Positive Solution ◽  
Stochastic Epidemic Model

In this paper, we present a differential infectivity SIR epidemic model with modified saturation incidences and stochastic perturbations. We show that the stochastic epidemic model has a unique global positive solution, and we utilize stochastic Lyapunov functions to show the asymptotic behavior of the solution.

Large Global Asymptotically Stability of an SEIR Epidemic Model with Distributed Time Delay

2011 ◽  
Vol 271-273 ◽  
pp. 428-434
Author(s):  
Jin Gao ◽  
Zhen Jin
Keyword(s):  
Time Delay ◽  
Equilibrium Point ◽  
State Space ◽  
Epidemic Model ◽  
Transmission Model ◽  
Force Of Infection ◽  
The Past ◽  
Distributed Time Delay ◽  
Globally Stable ◽  
Global Asymptotically Stability

An SEIR epidemic transmission model is formulated under the assumption that the force of infection at the present time depends on the number of infectives at the past. It is shown that a desease free equilibrium is globally stable if no epidemic equilibrium point exists. Further the epidemic equilibrium (if it exists) is globally stable in the who;e state space except the neighborhood of the desease free equilibrium.

scholarly journals Optimal Allocation of Vaccine and Antiviral Drugs for Influenza Containment over Delayed Multiscale Epidemic Model considering Time-Dependent Transmission Rate

2021 ◽  
Vol 2021 ◽  
pp. 1-27
Author(s):  
Zohreh Abbasi ◽  
Iman Zamani ◽  
Amir Hossein Amiri Mehra ◽  
Asier Ibeas ◽  
Mohsen Shafieirad
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Epidemic Model ◽  
Disease Transmission ◽  
Optimal Allocation ◽  
Transmission Rate ◽  
Antiviral Treatment ◽  
Infected Individual ◽  
Optimal Time ◽  
Control Input

In this study, two types of epidemiological models called “within host” and “between hosts” have been studied. The within-host model represents the innate immune response, and the between-hosts model signifies the SEIR (susceptible, exposed, infected, and recovered) epidemic model. The major contribution of this paper is to break the chain of infectious disease transmission by reducing the number of susceptible and infected people via transferring them to the recovered people group with vaccination and antiviral treatment, respectively. Both transfers are considered with time delay. In the first step, optimal control theory is applied to calculate the optimal final time to control the disease within a host’s body with a cost function. To this end, the vaccination that represents the effort that converts healthy cells into resistant-to-infection cells in the susceptible individual’s body is used as the first control input to vaccinate the susceptible individual against the disease. Moreover, the next control input (antiviral treatment) is applied to eradicate the concentrations of the virus and convert healthy cells into resistant-to-infection cells simultaneously in the infected person’s body to treat the infected individual. The calculated optimal time in the first step is considered as the delay of vaccination and antiviral treatment in the SEIR dynamic model. Using Pontryagin’s maximum principle in the second step, an optimal control strategy is also applied to an SEIR mathematical model with a nonlinear transmission rate and time delay, which is computed as optimal time in the first step. Numerical results are consistent with the analytical ones and corroborate our theoretical results.

Sign in / Sign up

Export Citation Format

Share Document