scholarly journals An Sir Epidemic Model on a Population with Random Network and Household Structure, and Several Types of Individuals

2012 ◽  
Vol 44 (1) ◽  
pp. 63-86 ◽  
Author(s):  
Frank Ball ◽  
David Sirl
Keyword(s):  
Random Graph ◽  
Epidemic Model ◽  
Strong Approximation ◽  
Random Network ◽  
Configuration Model ◽  
Sir Epidemic Model ◽  
Social Contacts ◽  
Sir Epidemic ◽  
Major Outbreak ◽  
Strong Approximation Theorem

We consider a stochastic SIR (susceptible → infective → removed) epidemic model with several types of individuals. Infectious individuals can make infectious contacts on two levels, within their own ‘household’ and with their neighbours in a random graph representing additional social contacts. This random graph is an extension of the well-known configuration model to allow for several types of individuals. We give a strong approximation theorem which leads to a threshold theorem for the epidemic model and a method for calculating the probability of a major outbreak given few initial infectives. A multitype analogue of a theorem of Ball, Sirl and Trapman (2009) heuristically motivates a method for calculating the expected size of such a major outbreak. We also consider vaccination and give some short numerical illustrations of our results.

scholarly journals An Sir Epidemic Model on a Population with Random Network and Household Structure, and Several Types of Individuals

2012 ◽  
Vol 44 (01) ◽  
pp. 63-86 ◽  
Author(s):  
Frank Ball ◽  
David Sirl
Keyword(s):  
Random Graph ◽  
Epidemic Model ◽  
Strong Approximation ◽  
Random Network ◽  
Configuration Model ◽  
Sir Epidemic Model ◽  
Social Contacts ◽  
Sir Epidemic ◽  
Major Outbreak ◽  
Strong Approximation Theorem

We consider a stochastic SIR (susceptible → infective → removed) epidemic model with several types of individuals. Infectious individuals can make infectious contacts on two levels, within their own ‘household’ and with their neighbours in a random graph representing additional social contacts. This random graph is an extension of the well-known configuration model to allow for several types of individuals. We give a strong approximation theorem which leads to a threshold theorem for the epidemic model and a method for calculating the probability of a major outbreak given few initial infectives. A multitype analogue of a theorem of Ball, Sirl and Trapman (2009) heuristically motivates a method for calculating the expected size of such a major outbreak. We also consider vaccination and give some short numerical illustrations of our results.

scholarly journals The extinction time of a subcritical branching process related to the SIR epidemic on a random graph

2015 ◽  
Vol 52 (04) ◽  
pp. 1195-1201 ◽  
Author(s):  
Peter Windridge
Keyword(s):  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Large Population ◽  
Extinction Time ◽  
Exponential Tail ◽  
Sir Epidemic Model ◽  
Multitype Branching Process ◽  
Type Distribution ◽  
Sir Epidemic

We give an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to∞). We only require a second moment for the offspring-type distribution featuring in our model.

scholarly journals The extinction time of a subcritical branching process related to the SIR epidemic on a random graph

2015 ◽  
Vol 52 (4) ◽  
pp. 1195-1201 ◽  
Author(s):  
Peter Windridge
Keyword(s):  
Random Graph ◽  
Epidemic Model ◽  
Branching Process ◽  
Large Population ◽  
Extinction Time ◽  
Exponential Tail ◽  
Sir Epidemic Model ◽  
Multitype Branching Process ◽  
Type Distribution ◽  
Sir Epidemic

We give an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to∞). We only require a second moment for the offspring-type distribution featuring in our model.

Analisis Titik Kesetimbangan Dan Kestabilan Penyebaran Penyakit Malaria di Distrik Manokwari Barat Berdasarkan Model Epidemik SIR

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Fandy Fandy ◽  
Andi Fajeriani Wyrasti ◽  
Tri Widjajanti
Keyword(s):  
Epidemic Model ◽  
Sir Epidemic Model ◽  
Endemic Diseases ◽  
Sir Epidemic

<em>Stability and equilibrium of malaria&rsquo;s epidemics in Manokwari Barat district based on SIR epidemic model will be discussed in this paper. The SIR epidemic model can be applied to make a model of endemic diseases like malaria. Based on this research, there are 2 types of the equilibrium of malaria&rsquo;s epidemics in Manokwari Barat District, endemic and non endemic point.</em>

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.

scholarly journals Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yakui Xue ◽  
Tiantian Li
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

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