scholarly journals On the Construction of Some Deterministic and Stochastic Non-Local SIR Models

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2103
Author(s):  
Giacomo Ascione
Keyword(s):  
Deterministic Model ◽  
Large Population ◽  
Epidemic Models ◽  
Fluid Limit ◽  
Sir Epidemic ◽  
Generalized Fractional Calculus ◽  
Stochastic Epidemic ◽  
Non Local ◽  
Stochastic Analog ◽  
Large Population Limit

Fractional-order epidemic models have become widely studied in the literature. Here, we consider the generalization of a simple SIR model in the context of generalized fractional calculus and we study the main features of such model. Moreover, we construct semi-Markov stochastic epidemic models by using time changed continuous time Markov chains, where the parent process is the stochastic analog of a simple SIR epidemic. In particular, we show that, differently from what happens in the classic case, the deterministic model does not coincide with the large population limit of the stochastic one. This loss of fluid limit is then stressed in terms of numerical examples.

scholarly journals Coupling of Two SIR Epidemic Models with Variable Susceptibilities and Infectivities

2007 ◽  
Vol 44 (01) ◽  
pp. 41-57 ◽  
Author(s):  
Peter Neal
Keyword(s):  
Size Distribution ◽  
Random Graph ◽  
Epidemic Model ◽  
Epidemic Models ◽  
Final Size ◽  
Sir Epidemic ◽  
Novel Approach ◽  
Stochastic Epidemic ◽  
Sir Epidemic Models ◽  
Variable Susceptibility

The variable generalised stochastic epidemic model, which allows for variability in both the susceptibilities and infectivities of individuals, is analysed. A very different epidemic model which exhibits variable susceptibility and infectivity is the random-graph epidemic model. A suitable coupling of the two epidemic models is derived which enables us to show that, whilst the epidemics are very different in appearance, they have the same asymptotic final size distribution. The coupling provides a novel approach to studying random-graph epidemic models.

scholarly journals Who is the infector? General multi-type epidemics and real-time susceptibility processes

2019 ◽  
Vol 51 (2) ◽  
pp. 606-631
Author(s):  
Tom Britton ◽  
Ka Yin Leung ◽  
Pieter Trapman
Keyword(s):  
Real Time ◽  
Random Graph ◽  
Branching Processes ◽  
Large Population ◽  
Graph Representation ◽  
Type Model ◽  
Random Weights ◽  
Stochastic Epidemic ◽  
Special Case ◽  
Large Population Limit

AbstractWe couple a multi-type stochastic epidemic process with a directed random graph, where edges have random weights (traversal times). This random graph representation is used to characterise the fractions of individuals infected by the different types of vertices among all infected individuals in the large population limit. For this characterisation, we rely on the theory of multi-type real-time branching processes. We identify a special case of the two-type model in which the fraction of individuals of a certain type infected by individuals of the same type is maximised among all two-type epidemics approximated by branching processes with the same mean offspring matrix.

scholarly journals Coupling of Two SIR Epidemic Models with Variable Susceptibilities and Infectivities

2007 ◽  
Vol 44 (1) ◽  
pp. 41-57 ◽  
Author(s):  
Peter Neal
Keyword(s):  
Size Distribution ◽  
Random Graph ◽  
Epidemic Model ◽  
Epidemic Models ◽  
Final Size ◽  
Sir Epidemic ◽  
Novel Approach ◽  
Stochastic Epidemic ◽  
Sir Epidemic Models ◽  
Variable Susceptibility

The variable generalised stochastic epidemic model, which allows for variability in both the susceptibilities and infectivities of individuals, is analysed. A very different epidemic model which exhibits variable susceptibility and infectivity is the random-graph epidemic model. A suitable coupling of the two epidemic models is derived which enables us to show that, whilst the epidemics are very different in appearance, they have the same asymptotic final size distribution. The coupling provides a novel approach to studying random-graph epidemic models.

scholarly journals Global dynamics of a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaodong Wang ◽  
Chunxia Wang ◽  
Kai Wang
Keyword(s):  
Vertical Transmission ◽  
Epidemic Model ◽  
Media Coverage ◽  
Existence And Uniqueness ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Global Dynamics ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Stochastic Sir

AbstractIn this paper, we study a novel deterministic and stochastic SIR epidemic model with vertical transmission and media coverage. For the deterministic model, we give the basic reproduction number $R_{0}$ R 0 which determines the extinction or prevalence of the disease. In addition, for the stochastic model, we prove existence and uniqueness of the positive solution, and extinction and persistence in mean. Furthermore, we give numerical simulations to verify our results.

On interarrival times in simple stochastic epidemic models

1982 ◽  
Vol 19 (4) ◽  
pp. 835-841
Author(s):  
Grace Yang ◽  
C. L. Chiang
Keyword(s):  
Probability Distributions ◽  
Asymptotic Distributions ◽  
Epidemic Models ◽  
Stochastic Epidemic Models ◽  
Stochastic Epidemic ◽  
Interarrival Times

The probability distributions of the size and the duration of simple stochastic epidemic models are well known. However, in most instances, the solutions are too complicated to be of practical use. In this note, interarrival times of the infectives are utilized to study asymptotic distributions of the duration of the epidemic for a class of simple epidemic models. A brief summary of the results on simple epidemic models in terms of interarrival times is included.

scholarly journals Global analysis of stochastic SIR model with variable diffusion rates

2018 ◽  
Vol 49 (2) ◽  
pp. 155-182 ◽  
Author(s):  
Pitchaimani M. ◽  
Rajasekar S.P.
Keyword(s):  
Equilibrium Solution ◽  
Deterministic Model ◽  
Global Analysis ◽  
Sir Epidemic Model ◽  
Treatment Rate ◽  
Stochastic Version ◽  
Sir Epidemic ◽  
Variable Diffusion ◽  
Stochastic Sir ◽  
Stochastic Sir Model

In this article, a stochastic SIR epidemic model with treatment rate in a population of varying size is proposed and investigated. For the stochastic version, we briefly discuss the existence of global unique solutions and using the Lyapunov function, the disease free equilibrium solution is globally asymptotic stabe if $\mathcal{R}_0\leq1$ and the endemic equilibrium solution is obtained when $\mathcal{R}_0>1$. The main attention is paid to the $p$th-moment exponentially stable on the system, proved under suitable assumptions on the white noise perturbations and the optimal control for the deterministic model. Finally numerical simulations are done to show our theoretical results and to demonstrate the complicated dynamics of the model.

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