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Exact Equations for SIR Epidemics on Tree Graphs.

Sharkey KJ, Kiss IZ, Wilkinson RR, Simon PL - Bull. Math. Biol. (2013)

Bottom Line: We consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected.We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph.Moreover, this "deterministic" representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK, kjs@liv.ac.uk.

ABSTRACT
We consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected. We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph. Moreover, this "deterministic" representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically.

No MeSH data available.


Related in: MedlinePlus

Shown is a state which cannot arise on a tree graph where there is only one initially infectious node
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Fig3: Shown is a state which cannot arise on a tree graph where there is only one initially infectious node

Mentions: When infection is initiated on a tree graph at a single individual, infection must always proceed in linear chains. Consequently there is no possibility of the state IkSjIi illustrated in Fig. 3 arising because an infection initiated at either k or i must pass through j to get to the other node. Furthermore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \langle S_jI_k\rangle=\langle S_iS_jI_k \rangle +\langle I_iS_jI_k\rangle+\langle R_iS_jI_k\rangle, \end{aligned}$$ \end{document} but since 〈IiSjIk〉=0 and consequently 〈RiSjIk〉=0, we have: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \langle S_jI_k\rangle=\langle S_iS_jI_k \rangle \end{aligned}$$ \end{document} reducing (1) to the following closed system: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \dot{\langle S_i\rangle} =&-\sum _{j=1}^P T_{ij}\langle S_iI_j\rangle , \\ \dot{\langle I_i\rangle} =&\sum _{j=1}^P T_{ij}\langle S_iI_j\rangle-\gamma_i\langle I_i\rangle, \\ \dot{\langle S_iI_j\rangle} =&\sum _{k=1, k\neq i}^P T_{jk}\langle S_jI_k\rangle-T_{ij}\langle S_iI_j\rangle-\gamma_j\langle S_iI_j\rangle, \\ \dot{\langle S_iS_j\rangle} =&-\sum _{k=1, k\neq j}^PT_{ik}\langle I_kS_i\rangle-\sum_{k=1,k\neq i}^PT_{jk} \langle S_jI_k\rangle. \end{aligned}$$ \end{document} Similar arguments show that this can be written in the form of (3). Fig. 3


Exact Equations for SIR Epidemics on Tree Graphs.

Sharkey KJ, Kiss IZ, Wilkinson RR, Simon PL - Bull. Math. Biol. (2013)

Shown is a state which cannot arise on a tree graph where there is only one initially infectious node
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4541714&req=5

Fig3: Shown is a state which cannot arise on a tree graph where there is only one initially infectious node
Mentions: When infection is initiated on a tree graph at a single individual, infection must always proceed in linear chains. Consequently there is no possibility of the state IkSjIi illustrated in Fig. 3 arising because an infection initiated at either k or i must pass through j to get to the other node. Furthermore, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \langle S_jI_k\rangle=\langle S_iS_jI_k \rangle +\langle I_iS_jI_k\rangle+\langle R_iS_jI_k\rangle, \end{aligned}$$ \end{document} but since 〈IiSjIk〉=0 and consequently 〈RiSjIk〉=0, we have: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \langle S_jI_k\rangle=\langle S_iS_jI_k \rangle \end{aligned}$$ \end{document} reducing (1) to the following closed system: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \dot{\langle S_i\rangle} =&-\sum _{j=1}^P T_{ij}\langle S_iI_j\rangle , \\ \dot{\langle I_i\rangle} =&\sum _{j=1}^P T_{ij}\langle S_iI_j\rangle-\gamma_i\langle I_i\rangle, \\ \dot{\langle S_iI_j\rangle} =&\sum _{k=1, k\neq i}^P T_{jk}\langle S_jI_k\rangle-T_{ij}\langle S_iI_j\rangle-\gamma_j\langle S_iI_j\rangle, \\ \dot{\langle S_iS_j\rangle} =&-\sum _{k=1, k\neq j}^PT_{ik}\langle I_kS_i\rangle-\sum_{k=1,k\neq i}^PT_{jk} \langle S_jI_k\rangle. \end{aligned}$$ \end{document} Similar arguments show that this can be written in the form of (3). Fig. 3

Bottom Line: We consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected.We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph.Moreover, this "deterministic" representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK, kjs@liv.ac.uk.

ABSTRACT
We consider Markovian susceptible-infectious-removed (SIR) dynamics on time-invariant weighted contact networks where the infection and removal processes are Poisson and where network links may be directed or undirected. We prove that a particular pair-based moment closure representation generates the expected infectious time series for networks with no cycles in the underlying graph. Moreover, this "deterministic" representation of the expected behaviour of a complex heterogeneous and finite Markovian system is straightforward to evaluate numerically.

No MeSH data available.


Related in: MedlinePlus