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An epidemic model on a network having two group structures with tunable overlap

Frank Ball, Tom Britton, Peter Neal

Abstract

A network epidemic model is studied. The underlying social network has two different types of group structures, households and workplaces, such that each individual belongs to exactly one household and one workplace. The random network is constructed such that a parameter $θ$ controls the degree of overlap between the two group structures: $θ=0$ corresponding to all household members belonging to the same workplace and $θ=1$ to all household members belonging to distinct workplaces. On the network a stochastic SIR epidemic is defined, having an arbitrary but specified infectious period distribution, with global (community), household and workplace infectious contacts. The stochastic epidemic model is analysed as the population size $n\to\infty$ with the (asymptotic) probability, and size, of a major outbreak obtained. These results are proved in greater generality than existing results in the literature by allowing for any fixed $0 \leq θ\leq 1$, a non-constant infectious period distribution, the presence or absence of global infection and potentially (asymptotically) infinite local outbreaks.

An epidemic model on a network having two group structures with tunable overlap

Abstract

A network epidemic model is studied. The underlying social network has two different types of group structures, households and workplaces, such that each individual belongs to exactly one household and one workplace. The random network is constructed such that a parameter controls the degree of overlap between the two group structures: corresponding to all household members belonging to the same workplace and to all household members belonging to distinct workplaces. On the network a stochastic SIR epidemic is defined, having an arbitrary but specified infectious period distribution, with global (community), household and workplace infectious contacts. The stochastic epidemic model is analysed as the population size with the (asymptotic) probability, and size, of a major outbreak obtained. These results are proved in greater generality than existing results in the literature by allowing for any fixed , a non-constant infectious period distribution, the presence or absence of global infection and potentially (asymptotically) infinite local outbreaks.

Paper Structure

This paper contains 25 sections, 8 theorems, 120 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The epidemic model with two group structures and tunable overlap
  3. The network model
  4. The epidemic model on the network
  5. Parametrization
  6. Notation
  7. Main Results
  8. Numerical illustrations
  9. Derivations and proofs when $\pi_G>0$
  10. Introduction
  11. General framework
  12. Local infectious clumps and susceptibility sets
  13. Complexes
  14. Local infectious clumps
  15. Major outbreak probability
  16. ...and 10 more sections

Key Result

Theorem 3.1

For the epidemic model with two group structures and tunable overlap defined in Section sec-model, When $\pi_G>0$, the constant $\rho$ (the major outbreak probability) is given by $\rho=1-\xi$, where $\xi$ is the smallest solution in $[0,1]$ of $\phi_A(\beta_G(1-s))=s$ and the Laplace transform $\phi_A(\nu)$ is given by equ:phiA in Section subsec-outbreakprob. The major outbreak probability $\rho

Figures (4)

  • Figure 1: Histograms of simulated fraction of the population infected for epidemics with $d=1$, $h=w=4$, $\beta=3$, $\pi_G=0.025$, $\pi_{H|G^c}=0.5$ and $I \equiv 1$. Each plot is based on 100,000 simulations. See text for further details.
  • Figure 2: Comparison of limiting and simulated estimates of the probability of a major outbreak $\rho$ (left column) and the fraction of the population infected by a major outbreak $z$ (right column) based on 10,000 simulations for each choice of $(n, d)$. The blue dashed horizontal lines depict the asymptotic values. The black crosses show the estimated values based on simulations, with the red vertical lines giving approximate $95\%$ confidence intervals. See text for further details.
  • Figure 3: Dependence of the limiting final size, $z$, on the probability an individual is a mover, $\theta$, for fixed $d=1,2,3,4$ when $h=3$, $\beta=3$, $\pi_G=0.025$ and $\pi_{H|G^c}=0.5$; in the left panel $I \equiv 1$ and in the right panel $I \sim {\rm Exp}(1)$.
  • Figure 4: Illustration of the construction of a population of $n=16$ individuals divided into 4 workplaces, labelled $A$, $B$, $C$, $D$, of size $4(=w)$ with each workplace consisting of $2(=d)$ households of size $2(=h)$. Red squares denote workplaces and green ellipses denote households. (a) The initial workplaces with solid circles representing remainers and open circles representing movers with movers numbered. There are 8 movers and 8 remainers. (b) The construction of complexes. Open squares represent the places created in a workplace by movers and the number in the square denotes the mover who fills the place. Note that in complexes A and D there are households consisting completely of movers with the household residing outside the workplace. (c) The final population structure created from the complexes.

Theorems & Definitions (17)

  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Remark 5.3
  • Remark 6.1
  • Lemma 6.2
  • ...and 7 more