Disease invasion on community networks with enviro

Authors: Joseph H Tien Zhisheng Shuai Marisa C Eisenberg P van den Driessche

Publish Date: 2014/05/05

Volume: 70, Issue: 5, Pages: 1065-1092

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Abstract

The ability of disease to invade a community network that is connected by environmental pathogen movement is examined Each community is modeled by a susceptible–infectious–recovered SIR framework that includes an environmental pathogen reservoir and the communities are connected by pathogen movement on a strongly connected weighted directed graph Disease invasibility is determined by the basic reproduction number mathcalR 0 for the domain The domain mathcalR 0 is computed through a Laurent series expansion with perturbation parameter corresponding to the ratio of the pathogen decay rate to the rate of water movement When movement is fast relative to decay mathcalR 0 is determined by the product of two weighted averages of the community characteristics The weights in these averages correspond to the network structure through the rooted spanning trees of the weighted directed graph Clustering of disease “hot spots” influences disease invasibility In particular clustering hot spots together according to a generalization of the group inverse of the Laplacian matrix facilitates disease invasionJHT and MCE acknowledge support from the National Science Foundation OCE1115881 and the Mathematical Biosciences Institute DMS0931642 The research of PvdD is partially supported through a Discovery Grant from the Natural Science and Engineering Research Council of Canada NSERC ZS acknowledges support from the University of Central Florida through a startup grant The authors are grateful to the anonymous reviewers for their thoughtful constructive comments This paper was improved by discussions at a Research in Teams meeting 13rit168 held at the Banff International Research StationFor the convenience of the reader we present some definitions and standard results from graph theory used throughout the paper Further information can be found in graph theory textbooks such as West 2001 We include also a statement of the matrix tree theorem for weighted directed graphs A proof can be found in Moon 1970A directed graph digraph mathcal G=VE consists of a set V=1 2 ldots n of vertices and a set E=Emathcal G of directed arcs ij from vertex i to vertex j A directed graph mathcal G is weighted if each arc ji is assigned a positive weight m ij the weighted directed graph is denoted as mathcal G M with nonnegative weight matrix M=m ij and m ij0 if and only if jiin Emathcal G For example the weight matrix M may correspond to the movement matrix in the community network and thus the weighted directed graph mathcal G M corresponds to the community network A directed graph is strongly connected if for any ordered pair of vertices there exists a directed path from one to the other A weighted directed graph mathcal G M is strongly connected if and only if the weight matrix M is irreducible Berman and Plemmons 1979A rooted spanning tree intree mathcal T is a subgraph of mathcal G on n vertices such that mathcal T is connected with no cycles and has a root vertex such that every directed path between a nonroot vertex and the root is oriented towards the root The weight of a rooted spanning tree is the product of all arcs in the rooted spanning tree To illustrate consider the star graph shown in Fig 5 with arc weights a from periphery to center and b from center to periphery Figure 5b shows the single spanning intree rooted at the center This tree possesses n arcs with weight a giving a tree weight of an Each peripheral vertex roots a single spanning intree shown in Fig 5c Trees rooted at the periphery have n1 arcs with weight a and a single arc with weight b giving a tree weight of an1bA direct consequence of the matrix tree theorem is a relationship between ker L and the rooted spanning trees of mathcalG This is pointed out for example in Lemma 21 of Guo et al 2006 Consider the vector of cofactors c = c 11 dots c nnT Then Lc i = det L = 0 for i=1dots n so c belongs to ker L The dimension of the nullspace of L is equal to the number of connected components of mathcalG so for strongly connected digraphs L has a one dimensional nullspace spanned by c For strongly connected digraphs there is at least one intree rooted at each vertex and thus the entries of c are all positive for digraphs with nonnegative arc weights Let u i = c ii / sum j=1n c jj and let u = u 1 dots u nT Then u provides a basis vector for ker L with all positive entries and normalized so that sum i=1n u i = 1A weighted directed graph mathcal G M is balanced if the net inflow is equal to the net outflow at each vertex ie for each i sum jnot =i m ij = sum jnot =im ji For balanced community networks both the row sums and column sums of L are equal to zero giving that all cofactors of L are equal This in turn implies that all entries of u are equal to 1/n for balanced networks

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