Generalization of Pairwise Models to non-Markovian Epidemics on Networks

Kiss, István Z.; Röst, Gergely; Vizi, Zsolt

doi:10.1103/physrevlett.115.078701

Cited by 87 publications

(107 citation statements)

References 29 publications

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“…In [19] reproduction numbers are precisely described in both cases: in context of mean-field model, we use the basic reproduction number R 0 , which is the expected lifetime of an I node multiplied by the number of newly generated I nodes per unit time. On the other hand, the pairwise reproduction number R p 0 is the expected lifetime of an S − I link multiplied by the number of newly generated S − I links per unit time.…”

Section: Proof It Is Clear That [Ss](t) Remains Nonnegative If Thementioning

confidence: 99%

“…We omit the detailed calculations here (see [19]). Note that while in compartmental models R 0 can be interpreted as the growth factor of subsequent generations of infected individuals in the initial phase of the epidemic, R p 0 in the pairwise model can intuitively be understood as the growth factor of subsequent generations of infected links.…”

Section: Proof It Is Clear That [Ss](t) Remains Nonnegative If Thementioning

confidence: 99%

“…The mean-field model for fixed recovery time (see [19]) can also be derived from (2.8b) using the same arguments.…”

Section: (B) Fixed Recovery Time σmentioning

confidence: 99%

“…The most popular node-level models are perhaps the degree-based or heterogeneous mean-field models. The pairwise models offer an explicit treatment of the epidemic process both at node and link level [15,19]. Such and other models of epidemic dynamics of networks (see [31] for a review) have led to a much better understanding of the role of contact heterogeneity, assortativity and clustering of contacts.…”

Section: Introductionmentioning

confidence: 99%

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Pairwise approximation for SIR -type network epidemics with non-Markovian recovery

2018

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We present the generalised mean-field and pairwise models for non-Markovian epidemics on networks with arbitrary recovery time distributions. First we consider a hyperbolic partial differential equation (PDE) system, where the population of infective nodes and links are structured by age since infection. We show that the PDE system can be reduced to a system of integro-differential equations, which is analysed analytically and numerically. We investigate the asymptotic behaviour of the generalised model and provide an implicit analytical expression involving the final epidemic size and pairwise reproduction number. As an illustration of the applicability of the general model, we recover known results for the exponentially distributed and fixed recovery time cases. For gamma-and uniformly-distributed infectious periods, new pairwise models are derived. Theoretical findings are confirmed by comparing results from the new pairwise model and explicit stochastic network simulation. A major benefit of the generalised pairwise model lies in approximating the time evolution of the epidemic.

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Section: Proof It Is Clear That [Ss](t) Remains Nonnegative If Thementioning

confidence: 99%

Section: Proof It Is Clear That [Ss](t) Remains Nonnegative If Thementioning

confidence: 99%

“…The mean-field model for fixed recovery time (see [19]) can also be derived from (2.8b) using the same arguments.…”

Section: (B) Fixed Recovery Time σmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Pairwise approximation for SIR -type network epidemics with non-Markovian recovery

2018

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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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Semi‐Markov models of epidemics over networks with time delays

Ghousein

Moulay

Coirault

2022

Intl J Robust & Nonlinear

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In this work, we extend the Markov models describing the susceptible-infectedsusceptible (SIS) epidemics over undirected networks to take into account the virus minimum incubation period and the minimum recovery period of an infected individual. We represent both periods as time delays in the states of the extended model. However, due to the addition of time delays, the process loses its Markovain property. We use the generalized semi-Markov theory to introduce both the incubation and recovery delays to the probabilistic dynamical models. Hence, in this paper, we propose a time-delay version of the two principal models of the SIS epidemics over undirected networks: the exact 2 𝑁 −state model and the approximated 𝑁−intertwined model. Finally, using Lyapunov analysis, we give sufficient conditions that guarantee the global exponential stability of the time-delay 𝑁−intertwined model.

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Generalization of Pairwise Models to non-Markovian Epidemics on Networks

Cited by 87 publications

References 29 publications

Pairwise approximation for SIR -type network epidemics with non-Markovian recovery

Pairwise approximation for SIR -type network epidemics with non-Markovian recovery

Optimal allocation of resources for suppressing epidemic spreading on networks

Semi‐Markov models of epidemics over networks with time delays

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