TY - JOUR

T1 - Epidemics on networks

T2 - Reducing disease transmission using health emergency declarations and peer communication

AU - Azizi, Asma

AU - Montalvo, Cesar

AU - Espinoza, Baltazar

AU - Kang, Yun

AU - Castillo-Chavez, Carlos

N1 - Funding Information: This work was supported by the grant from the National Security Agency (NSAGrant H98230-J8-1-0005 ), National Science Foundation (NSF Grant 1716802 ), and James S. McDonnell Foundation ( 220020472 ). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Appendix In this appendix, first we define some models for the network structures we used for our simulation, then we present the pseudocode of our model. Definition 4.1 Erdős-Rényi random network Suppose we have n disconnected nodes, and then we make edge between two arbitrary nodes with probability p independent from every other edge. The graph constructed via this model is called Erdős-Rényi random network G ( n , p ) . The parameter p in the model is a weighting function p ∈ [ 0,1 ] , where for p closer to one the generated graph is more likely to construct graphs with more edges ( ERDdS & R&WI, 1959 ). Definition 4.2 Small-world network Suppose we have n nodes over a ring, where each node in the ring is connected to its k nearest neighbors in the ring. Then with probability p and independent from any other pair of edges we select two edges i 1 j 1 and i 2 j 2 such that j k is from the k nearest neighbor of i k for k = 1,2 , and rewire them. Then the constructed network is called Watts-Strogatz Smal-world network ( Watts & Strogatz, 1998 ) G ( n , k , p ) . Definition 4.3 Preferential attachment Scale-free network Suppose we want to generate a network of n nodes. We start with one node and follow the following procedure n times: the probability of attaching a new node to the existing ones is proportional to their current degree: P ( connect a new node i to existing node j ) = d e g ( j ) ∑ k ( d e g ( k ) ) , where the d e g ( k ) is the number of neighbors for node k, then with this probability we make new edge i j . The constructed network via this model is called Barabasi – Albert Scale-free network ( Barabási & Albert, 1999 ) . The degree distribution of this network follows a power-law distribution. Now, we define assumptions and algorithm of our model Table 2 . Person i at any given day t has one of the states: S u , S a , S i , I q , I f , or R . The fraction of infected people at dat t is P ( t ) , and P * is a positive fraction ∈ [ 0,1 ] . We define t * = m i n t { P ( t ) = P * } which is the first time that P ( t ) reaches P * . We assume Education level for each person k , x k does not change by time. Also prior to time t * every susceptible person is at state S u , and finally the time scale for each update is day, in which each person can have at most one update. The following algorithms briefly shows some key part of the model. Table 2 Table of notation for a conventional network G in algorithms. Appendix Table 2 Notation Description S t Set of susceptible nodes at time t U t Set of unaware nodes at time t A t Set of aware nodes at time t C t Set of careless nodes at time t I t Set of infected nodes at time t Q t Set of quarantine nodes at time t F t Set of free nodes at time t R t Set of recovered nodes at time t G . N ( k ) Set of neighbors of node k in Network G I P (k) Infection period for infected node k A P (k) Awareness period for aware node k e r n ( α ) Exponential random number with average α for α > 0 u r n Uniform random number in [ 0,1 ] c k j Probability of having contact between two neighbors k and j A k → B Element k moves from set A to set B Algorithm 1 The probability of contact per day between two neighbors: infected k and susceptible j. σ k ˆ = 1 | k ∈ Q t ; σ j = κ x j ( θ + x j ) ( 1 + p r ( t ) ) | j ∈ A t + 1 | j ∉ A t ; Σ k j = def σ k ˆ σ j Algorithm 2 Dynamics within a typical day t Image 1 Funding Information: This work was supported by the grant from the National Security Agency (NSAGrantH98230-J8-1-0005), National Science Foundation(NSF Grant 1716802), and James S. McDonnell Foundation (220020472). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Publisher Copyright: © 2019 The Authors

PY - 2020

Y1 - 2020

N2 - Understanding individual decisions in a world where communications and information move instantly via cell phones and the internet, contributes to the development and implementation of policies aimed at stopping or ameliorating the spread of diseases. In this manuscript, the role of official social network perturbations generated by public health officials to slow down or stop a disease outbreak are studied over distinct classes of static social networks. The dynamics are stochastic in nature with individuals (nodes) being assigned fixed levels of education or wealth. Nodes may change their epidemiological status from susceptible, to infected and to recovered. Most importantly, it is assumed that when the prevalence reaches a pre-determined threshold level, P*, information, called awareness in our framework, starts to spread, a process triggered by public health authorities. Information is assumed to spread over the same static network and whether or not one becomes a temporary informer, is a function of his/her level of education or wealth and epidemiological status. Stochastic simulations show that threshold selection P* and the value of the average basic reproduction number impact the final epidemic size differentially. For the Erdős-Rényi and Small-world networks, an optimal choice for P* that minimize the final epidemic size can be identified under some conditions while for Scale-free networks this is not case.

AB - Understanding individual decisions in a world where communications and information move instantly via cell phones and the internet, contributes to the development and implementation of policies aimed at stopping or ameliorating the spread of diseases. In this manuscript, the role of official social network perturbations generated by public health officials to slow down or stop a disease outbreak are studied over distinct classes of static social networks. The dynamics are stochastic in nature with individuals (nodes) being assigned fixed levels of education or wealth. Nodes may change their epidemiological status from susceptible, to infected and to recovered. Most importantly, it is assumed that when the prevalence reaches a pre-determined threshold level, P*, information, called awareness in our framework, starts to spread, a process triggered by public health authorities. Information is assumed to spread over the same static network and whether or not one becomes a temporary informer, is a function of his/her level of education or wealth and epidemiological status. Stochastic simulations show that threshold selection P* and the value of the average basic reproduction number impact the final epidemic size differentially. For the Erdős-Rényi and Small-world networks, an optimal choice for P* that minimize the final epidemic size can be identified under some conditions while for Scale-free networks this is not case.

KW - Awareness spread

KW - Behavior change

KW - Erdős-rényi network

KW - Outbreak and epidemic threats

KW - Scale-free network

KW - Small-world network

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U2 - 10.1016/j.idm.2019.11.002

DO - 10.1016/j.idm.2019.11.002

M3 - Article

SN - 2468-0427

VL - 5

SP - 12

EP - 22

JO - Infectious Disease Modelling

JF - Infectious Disease Modelling

ER -