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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

Open triple graph
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Fig2: Open triple graph

Mentions: In this section we introduce a new notation which will assist in formulating the set of differential equations for the full system. In (1) we formulated the differential equations up to the level of pairs and we said that this could be continued up to the full system level. This will be done formally here. In order to make the method clearer, using our existing notation let us first evaluate the full set of equations for the undirected line graph with three nodes which we refer to as the open triple, depicted in Fig. 2. Here we shall assume that the transmission rate parameter is τ across both links and that the removal rate is γ for all three nodes. Firstly we write all of the single node equations for this network. From (1): 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \begin{aligned} \dot{\langle I_1\rangle}&= \tau\langle S_1I_2\rangle-\gamma\langle I_1\rangle, \\ \dot{\langle I_2\rangle}&= \tau\langle I_1S_2\rangle+\tau\langle S_2I_3 \rangle-\gamma\langle I_2\rangle, \\ \dot{\langle I_3\rangle}&= \tau\langle I_2S_3 \rangle-\gamma\langle I_3\rangle, \end{aligned} \end{aligned}$$ \end{document} and 5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \begin{aligned} \dot{\langle S_1\rangle}&= -\tau\langle S_1I_2\rangle, \\ \dot{\langle S_2\rangle}&= -\tau\langle I_1S_2\rangle-\tau\langle S_2I_3 \rangle, \\ \dot{\langle S_3\rangle}&= -\tau\langle I_2S_3 \rangle. \end{aligned} \end{aligned}$$ \end{document} We also need to specify the following equations for pairs: 6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \begin{aligned} \dot{\langle S_1I_2\rangle}&= \tau\langle S_1S_2I_3\rangle- \tau \langle S_1I_2\rangle-\gamma\langle S_1I_2 \rangle, \\ \dot{\langle I_1S_2\rangle}&= -\tau\langle I_1S_2I_3\rangle-\tau \langle I_1S_2\rangle-\gamma\langle I_1S_2 \rangle, \\ \dot{\langle S_2I_3\rangle}&= -\tau\langle I_1S_2I_3\rangle-\tau \langle S_2I_3\rangle-\gamma\langle S_2I_3 \rangle, \\ \dot{\langle I_2S_3\rangle}&= \tau\langle I_1S_2S_3\rangle-\tau \langle I_2S_3\rangle-\gamma\langle I_2S_3 \rangle. \end{aligned} \end{aligned}$$ \end{document} Finally, at the triple level we have from the master equation (since the system has only three nodes): 7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \begin{aligned} \dot{\langle S_1S_2I_3 \rangle}&= -\tau\langle S_1S_2I_3\rangle - \gamma\langle S_1S_2I_3\rangle, \\ \dot{\langle I_1S_2I_3 \rangle}&= -2\tau\langle I_1S_2I_3\rangle -2\gamma\langle I_1S_2I_3\rangle, \\ \dot{\langle I_1S_2S_3\rangle}&= -\tau \langle I_1S_2S_3\rangle- \gamma\langle I_1S_2S_3\rangle. \end{aligned} \end{aligned}$$ \end{document}Fig. 2


Exact Equations for SIR Epidemics on Tree Graphs.

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

Open triple graph
© Copyright Policy
Related In: Results  -  Collection

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

Fig2: Open triple graph
Mentions: In this section we introduce a new notation which will assist in formulating the set of differential equations for the full system. In (1) we formulated the differential equations up to the level of pairs and we said that this could be continued up to the full system level. This will be done formally here. In order to make the method clearer, using our existing notation let us first evaluate the full set of equations for the undirected line graph with three nodes which we refer to as the open triple, depicted in Fig. 2. Here we shall assume that the transmission rate parameter is τ across both links and that the removal rate is γ for all three nodes. Firstly we write all of the single node equations for this network. From (1): 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \begin{aligned} \dot{\langle I_1\rangle}&= \tau\langle S_1I_2\rangle-\gamma\langle I_1\rangle, \\ \dot{\langle I_2\rangle}&= \tau\langle I_1S_2\rangle+\tau\langle S_2I_3 \rangle-\gamma\langle I_2\rangle, \\ \dot{\langle I_3\rangle}&= \tau\langle I_2S_3 \rangle-\gamma\langle I_3\rangle, \end{aligned} \end{aligned}$$ \end{document} and 5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \begin{aligned} \dot{\langle S_1\rangle}&= -\tau\langle S_1I_2\rangle, \\ \dot{\langle S_2\rangle}&= -\tau\langle I_1S_2\rangle-\tau\langle S_2I_3 \rangle, \\ \dot{\langle S_3\rangle}&= -\tau\langle I_2S_3 \rangle. \end{aligned} \end{aligned}$$ \end{document} We also need to specify the following equations for pairs: 6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \begin{aligned} \dot{\langle S_1I_2\rangle}&= \tau\langle S_1S_2I_3\rangle- \tau \langle S_1I_2\rangle-\gamma\langle S_1I_2 \rangle, \\ \dot{\langle I_1S_2\rangle}&= -\tau\langle I_1S_2I_3\rangle-\tau \langle I_1S_2\rangle-\gamma\langle I_1S_2 \rangle, \\ \dot{\langle S_2I_3\rangle}&= -\tau\langle I_1S_2I_3\rangle-\tau \langle S_2I_3\rangle-\gamma\langle S_2I_3 \rangle, \\ \dot{\langle I_2S_3\rangle}&= \tau\langle I_1S_2S_3\rangle-\tau \langle I_2S_3\rangle-\gamma\langle I_2S_3 \rangle. \end{aligned} \end{aligned}$$ \end{document} Finally, at the triple level we have from the master equation (since the system has only three nodes): 7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \begin{aligned} \dot{\langle S_1S_2I_3 \rangle}&= -\tau\langle S_1S_2I_3\rangle - \gamma\langle S_1S_2I_3\rangle, \\ \dot{\langle I_1S_2I_3 \rangle}&= -2\tau\langle I_1S_2I_3\rangle -2\gamma\langle I_1S_2I_3\rangle, \\ \dot{\langle I_1S_2S_3\rangle}&= -\tau \langle I_1S_2S_3\rangle- \gamma\langle I_1S_2S_3\rangle. \end{aligned} \end{aligned}$$ \end{document}Fig. 2

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