Exact deterministic representation of Markovian $${ SIR}$$ S I R epidemics on networks with and without loops

Kiss, István Z.; Morris, Charles G.; Sélley, Fanni M.; Šimon, Peter; Wilkinson, Robert R.

doi:10.1007/s00285-014-0772-0

Cited by 37 publications

(53 citation statements)

References 19 publications

(47 reference statements)

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“…(8) [6,8]. It is clear that the number of equations will be of the order of the number of directed links in the network, i.e., |A(D)|.…”

Section: Equations For Poisson Contact Processesmentioning

confidence: 99%

“…(3) and (7)] and the pair-based system (8) are equivalent in the sense that they produce the same time series for the probabilities of the states of individuals. Here we shall consider both systems for exponential and fixed recovery processes.…”

Section: Equations For Poisson Contact Processesmentioning

confidence: 99%

“…However, we also get equality in (20) whenever there are no directed cycles in the digraph D. Therefore, sufficient requirements for the exactness of Eqs. (2) to (8) are that (1) there is no more than one simple directed path from any individual to any other individual in D and (2) there are no directed cycles in D. Equations (1), (19), and (20) imply that…”

Section: Sir Dynamics On Non-tree Networkmentioning

confidence: 99%

“…(12) and (13) formalisms [all occurrences of H , S, I and R in Eqs. (1) to (8) are changed, respectively, to F , X,Y and Z, indicating inexactness]. Since they are implied by the message passing formalism, the solution of the pair-based equations (8) on non-tree networks, i.e., arbitrary digraphs, provides a rigorous lower bound on S i and approximations for I i , R i ∀ i ∈ V .…”

Section: Sir Dynamics On Non-tree Networkmentioning

confidence: 99%

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Message passing and moment closure for susceptible-infected-recovered epidemics on finite networks

Wilkinson

Sharkey

2014

Phys. Rev. E

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The message passing approach of Karrer and Newman [Phys. Rev. E 82, 016101 (2010)] is an exact and practicable representation of susceptible-infected-recovered dynamics on finite trees. Here we show that, assuming Poisson contact processes, a pair-based moment-closure representation [Sharkey, J. Math. Biol. 57, 311 (2008)] can be derived from their equations. We extend the applicability of both representations and discuss their relative merits. On arbitrary time-independent networks, as was shown for the message passing formalism, the pair-based moment-closure equations also provide a rigorous lower bound on the expected number of susceptibles at all times.

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“…(8) [6,8]. It is clear that the number of equations will be of the order of the number of directed links in the network, i.e., |A(D)|.…”

Section: Equations For Poisson Contact Processesmentioning

confidence: 99%

Section: Equations For Poisson Contact Processesmentioning

confidence: 99%

Section: Sir Dynamics On Non-tree Networkmentioning

confidence: 99%

Section: Sir Dynamics On Non-tree Networkmentioning

confidence: 99%

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Message passing and moment closure for susceptible-infected-recovered epidemics on finite networks

Wilkinson

Sharkey

2014

Phys. Rev. E

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“…An interesting generalization is how to solve the present constraint optimization problem based on other existing theoretical methods, such as pair mean-field method that takes into account the role of dynamical correlations between neighboring nodes [31][32][33][34][35][36][37][38][39]. Moreover, the method presented here could be applied to a number of other optimization problems, for example, controlling opinion dynamics in social networks [40].…”

mentioning

confidence: 99%

Optimal allocation of resources for suppressing epidemic spreading on networks

Chen

Zhang

et al. 2017

Phys. Rev. E

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Efficient allocation of limited medical resources is crucial for controlling epidemic spreading on networks. Based on the susceptible-infected-susceptible model, we solve an optimization problem as how best to allocate the limited resources so as to minimize the prevalence, providing that the curing rate of each node is positively correlated to its medical resource. By quenched meanfield theory and heterogeneous mean-field (HMF) theory, we prove that epidemic outbreak will be suppressed to the greatest extent if the curing rate of each node is directly proportional to its degree, under which the effective infection rate λ has a maximal threshold λ opt c = 1/ k where k is average degree of the underlying network. For weak infection region (λ λ opt c ), we combine a perturbation theory with Lagrange multiplier method (LMM) to derive the analytical expression of optimal allocation of the curing rates and the corresponding minimized prevalence. For general infection region (λ > λ opt c ), the high-dimensional optimization problem is converted into numerically solving low-dimensional nonlinear equations by the HMF theory and LMM. Counterintuitively, in the strong infection region the low-degree nodes should be allocated more medical resources than the high-degree nodes to minimize the prevalence. Finally, we use simulated annealing to validate the theoretical results. A challenging problem in epidemiology is how best to allocate limited resources of treatment and vaccination so that they will be most effective in suppressing or reducing outbreaks of epidemics. This problem has been a subject of intense research in statistical physics and many other disciplines [1,2]. Inspired by the percolation theory, the simplest strategy is to randomly choose a fraction of nodes to immunize. However, the random immunization is inefficient for heterogeneous networks. Later on, many more effective immunization strategies have been developed, ranging from global strategies like targeted immunization based on node degree [3] or betweenness centrality [4] to local strategies, like acquaintance immunization [5] and (bias) random walk immunization [6,7] and to some others in between [8]. Further improvements were done by graph partitioning [9] and the optimization of the susceptible size [10]. Besides the degree heterogeneity, community structure has also a major impact on disease immunity [11,12]. Recently, a message-passing approach was used to find an optimal set of nodes for immunization [13]. The immunization has been mapped onto the optimal percolation problem [14]. Based on the idea of explosive percolation, an "explosive immunization" method has been proposed [15]. However, some diseases like the common cold and influenza that can be modeled by the susceptible-infected-susceptible (SIS) model, do not confer immunity and individuals can be infected over and over again. Under the situations, one * Electronic address: chenhshf@ahu.edu.cn † Electronic address: hzhlj@ustc.edu.cn way to control the spread of the diseases is to reduce t...

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