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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

Star graph with P=4 nodes
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Fig4: Star graph with P=4 nodes

Mentions: We now consider the case of the undirected star graph with P=4 shown in Fig. 4, where again we assume that the strength is the same across each network link and is denoted by τ and the removal rate for each node is γ. Writing down the equations of the extended state space, there are two types of closure which need to be proved: one for the S−S−I triples and one for the I−S−I triples (see (1)). The graph has three triples ((1,4,3),(2,4,3),(1,4,2)), but it is sufficient to prove exactness for one of them. Hence we want to prove the following two relations: 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \begin{aligned} \langle S_4\rangle \langle S_1I_3S_4 \rangle&= \langle S_1S_4\rangle \langle I_3S_4\rangle, \\ \langle S_4\rangle \langle I_1I_3S_4 \rangle&= \langle I_1S_4\rangle \langle I_3S_4\rangle. \end{aligned} \end{aligned}$$ \end{document} For brevity, we adopt the alternative notation: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \bigl\langle \psi_4^S\bigr\rangle \bigl\langle \psi_{1,3,4}^{SIS}\bigr\rangle -\bigl\langle \psi_{1,4}^{SS} \bigr\rangle \bigl\langle \psi_{3,4}^{IS}\bigr\rangle =&0, \\ \bigl\langle \psi_4^S\bigr\rangle \bigl\langle _{1,3,4}^{IIS}\bigr\rangle -\bigl\langle \psi _{1,4}^{IS}\bigr\rangle \bigl\langle \psi_{3,4}^{IS} \bigr\rangle =&0. \end{aligned}$$ \end{document} We introduce \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \alpha_1=\bigl\langle \psi_4^S\bigr\rangle \bigl\langle \psi_{1,3,4}^{SIS}\bigr\rangle -\bigl\langle \psi_{1,4}^{SS}\bigr\rangle \bigl\langle \psi_{3,4}^{IS} \bigr\rangle . \end{aligned}$$ \end{document} By differentiating this, substituting in from the process equations and grouping terms, we obtain 13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \dot{\alpha}_1=-(\tau+\gamma)\alpha_1-\tau \alpha_2-\tau\alpha _3-\tau\alpha_4, \end{aligned}$$ \end{document} where: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \alpha_2 =&\bigl\langle \psi_{1,4}^{IS} \bigr\rangle \bigl\langle \psi _{1,3,4}^{SIS}\bigr\rangle -\bigl\langle \psi_{1,4}^{SS}\bigr\rangle \bigl\langle \psi _{1,3,4}^{IIS}\bigr\rangle , \\ \alpha_3 =&\bigl\langle \psi_{2,4}^{IS} \bigr\rangle \bigl\langle \psi _{1,3,4}^{SIS}\bigr\rangle -\bigl\langle \psi_{1,4}^{SS}\bigr\rangle \bigl\langle \psi _{2,3,4}^{IIS}\bigr\rangle , \\ \alpha_4 =&\bigl\langle \psi_4^S \bigr\rangle \bigl\langle \psi _{1,2,3,4}^{SIIS}\bigr\rangle -\bigl\langle \psi_{1,2,4}^{SIS}\bigr\rangle \bigl\langle \psi_{3,4}^{IS}\bigr\rangle . \end{aligned}$$ \end{document} Differentiating α2 we get 14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \dot{\alpha}_2=-2(\tau+\gamma)\alpha_2-\tau \alpha_5-\tau\alpha_6, \end{aligned}$$ \end{document} where: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \alpha_5 =&\bigl\langle \psi_{1,2,4}^{IIS} \bigr\rangle \bigl\langle \psi _{1,3,4}^{SIS}\bigr\rangle -\bigl\langle \psi_{1,2,4}^{SIS}\bigr\rangle \bigl\langle \psi _{1,3,4}^{IIS}\bigr\rangle , \\ \alpha_6 =&\bigl\langle \psi_{1,4}^{IS} \bigr\rangle \bigl\langle \psi _{1,2,3,4}^{SIIS}\bigr\rangle -\bigl\langle \psi_{1,4}^{SS}\bigr\rangle \bigl\langle \psi _{1,2,3,4}^{IIIS}\bigr\rangle . \end{aligned}$$ \end{document} The derivatives of α3 and α4 can be obtained similarly. Fig. 4


Exact Equations for SIR Epidemics on Tree Graphs.

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

Star graph with P=4 nodes
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Related In: Results  -  Collection

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Fig4: Star graph with P=4 nodes
Mentions: We now consider the case of the undirected star graph with P=4 shown in Fig. 4, where again we assume that the strength is the same across each network link and is denoted by τ and the removal rate for each node is γ. Writing down the equations of the extended state space, there are two types of closure which need to be proved: one for the S−S−I triples and one for the I−S−I triples (see (1)). The graph has three triples ((1,4,3),(2,4,3),(1,4,2)), but it is sufficient to prove exactness for one of them. Hence we want to prove the following two relations: 12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \begin{aligned} \langle S_4\rangle \langle S_1I_3S_4 \rangle&= \langle S_1S_4\rangle \langle I_3S_4\rangle, \\ \langle S_4\rangle \langle I_1I_3S_4 \rangle&= \langle I_1S_4\rangle \langle I_3S_4\rangle. \end{aligned} \end{aligned}$$ \end{document} For brevity, we adopt the alternative notation: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \bigl\langle \psi_4^S\bigr\rangle \bigl\langle \psi_{1,3,4}^{SIS}\bigr\rangle -\bigl\langle \psi_{1,4}^{SS} \bigr\rangle \bigl\langle \psi_{3,4}^{IS}\bigr\rangle =&0, \\ \bigl\langle \psi_4^S\bigr\rangle \bigl\langle _{1,3,4}^{IIS}\bigr\rangle -\bigl\langle \psi _{1,4}^{IS}\bigr\rangle \bigl\langle \psi_{3,4}^{IS} \bigr\rangle =&0. \end{aligned}$$ \end{document} We introduce \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \alpha_1=\bigl\langle \psi_4^S\bigr\rangle \bigl\langle \psi_{1,3,4}^{SIS}\bigr\rangle -\bigl\langle \psi_{1,4}^{SS}\bigr\rangle \bigl\langle \psi_{3,4}^{IS} \bigr\rangle . \end{aligned}$$ \end{document} By differentiating this, substituting in from the process equations and grouping terms, we obtain 13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \dot{\alpha}_1=-(\tau+\gamma)\alpha_1-\tau \alpha_2-\tau\alpha _3-\tau\alpha_4, \end{aligned}$$ \end{document} where: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \alpha_2 =&\bigl\langle \psi_{1,4}^{IS} \bigr\rangle \bigl\langle \psi _{1,3,4}^{SIS}\bigr\rangle -\bigl\langle \psi_{1,4}^{SS}\bigr\rangle \bigl\langle \psi _{1,3,4}^{IIS}\bigr\rangle , \\ \alpha_3 =&\bigl\langle \psi_{2,4}^{IS} \bigr\rangle \bigl\langle \psi _{1,3,4}^{SIS}\bigr\rangle -\bigl\langle \psi_{1,4}^{SS}\bigr\rangle \bigl\langle \psi _{2,3,4}^{IIS}\bigr\rangle , \\ \alpha_4 =&\bigl\langle \psi_4^S \bigr\rangle \bigl\langle \psi _{1,2,3,4}^{SIIS}\bigr\rangle -\bigl\langle \psi_{1,2,4}^{SIS}\bigr\rangle \bigl\langle \psi_{3,4}^{IS}\bigr\rangle . \end{aligned}$$ \end{document} Differentiating α2 we get 14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \dot{\alpha}_2=-2(\tau+\gamma)\alpha_2-\tau \alpha_5-\tau\alpha_6, \end{aligned}$$ \end{document} where: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \alpha_5 =&\bigl\langle \psi_{1,2,4}^{IIS} \bigr\rangle \bigl\langle \psi _{1,3,4}^{SIS}\bigr\rangle -\bigl\langle \psi_{1,2,4}^{SIS}\bigr\rangle \bigl\langle \psi _{1,3,4}^{IIS}\bigr\rangle , \\ \alpha_6 =&\bigl\langle \psi_{1,4}^{IS} \bigr\rangle \bigl\langle \psi _{1,2,3,4}^{SIIS}\bigr\rangle -\bigl\langle \psi_{1,4}^{SS}\bigr\rangle \bigl\langle \psi _{1,2,3,4}^{IIIS}\bigr\rangle . \end{aligned}$$ \end{document} The derivatives of α3 and α4 can be obtained similarly. Fig. 4

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