Complexity

Volume 2018 (2018), Article ID 5024327, 9 pages

https://doi.org/10.1155/2018/5024327

## Epidemic Spreading in Complex Networks with Resilient Nodes: Applications to FMD

Correspondence should be addressed to Chang Hyeong Lee; rk.ca.tsinu@eelhc

Received 10 September 2017; Revised 30 January 2018; Accepted 13 February 2018; Published 15 March 2018

Academic Editor: Roberto Natella

Copyright © 2018 Pilwon Kim and Chang Hyeong Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

At the outbreak of the animal epidemic disease, farms that recover quickly from partially infected state can delay or even suppress the wide spreading of the infection over farm networks. In this work, we focus on how the spatial transmission of the infection is affected by both factors, the topology of networks and the internal resilience mechanism of nodes. We first develop an individual farm model to examine the influence of initial number of infected individuals and vaccination rate on the transmission in a single farm. Based on such intrafarm model, the farm network is constructed which reflects disease transmission between farms at various stages. We explore the impact of the farms vaccinated at low rates on the disease transmission into entire farm network and investigate the effect of the control on hub farms on the transmission over the farm network. It is shown that intensive control on the farms vaccinated at low rates and hub farms effectively reduces the potential risk of foot-and-mouth disease (FMD) outbreak on the farm network.

#### 1. Introduction

The epidemic spreading in networks has attracted growing attention in recent years [1–3]. One of the main reasons for studying the spread process in networks is reproducing the actual dynamics of the disease and finding an effective control strategy to eradicate the infection. The spreading of infectious animal disease between farms exhibits the vulnerability of the community structure, which has been mostly investigated in terms of the network topology [4–6]. However, whether or not the infection is suppressed before gaining “momentum” to spread over networks also depends on the nodes’ responses to the epidemic [7]. If each farm at a node is vaccinated at a high level and can recover from accidental and partial infection, such resilient mechanism can delay or even suppress the wide spreading of the infection.

Among animal infectious diseases, foot-and-mouth disease (FMD) is an economically important disease. Mathematical models have been developed to simulate the possible progress of FMD transmission under various scenarios, so that relevant researchers and agents can use them for implementing control policies to prevent and reduce the risk of the disease outbreak. A discretized deterministic SLIR model, where , , , and denote the susceptible, the latent, the infectious, and the recovered states, respectively, was developed to describe FMD transmission in a farm [8]. An SVLI model, where denotes the vaccinated state, was considered to investigate FMD transmission in a farm [9]. The authors investigated the stability of the disease-free equilibrium and performed simulations to illustrate the impact of vaccination and culling on controlling the disease in a farm. Concerning FMD transmission between farms, Keeling et al., using their developed individual farm-based stochastic model, found that the spatial distribution and size of farms have a certain effect on the pattern and regional variability of FMD outbreaks [10] and investigated the effect of vaccination and culling on control of FMD transmission [11]. Moreover, optimal reactive vaccination strategies were investigated in [12] and various vaccination strategies were studied for stochastic SIR model on a random network of social contacts with household structure [13]. Farm-to-farm contacts due to movements of operators and vehicles have long been acknowledged as a relevant factor in disease transmission in livestock systems [14–16]. It is observed that the spread of infections can be tremendously strengthened on such contact networks. Particularly, some sorts of complex networks that connect susceptible/infected nodes are prone to the epidemic spreading with the persistence of infections regardless of the spreading rates [17, 18]. The bond percolation on random networks has been used to derive the basic epidemiological quantities and to predict disease transmission. A brief overview of the compartment SIR model in different type of the contact network can be found in [19, 20] and a comprehensive review for modeling FMD transmission in and between farms is done in [21].

In this work, we want to find how much the mutual dependence of farms weakens their resistance to external infections. With intensive and accumulative livestock farming in the industrialized world, farms are more densely connected even in the distance. Farms that are closely linked and share the common resources are generally more vulnerable to disease than an isolated farm. We are going to derive the minimum vaccination level to protect a farm community from sporadic exposure to external infectious source. One of the key problems in the epidemiology is how to effectively control the transmission of infectious disease by immunization of the population. The most undesirable situation is internalization or localization of the disease even under regular practice of vaccination. This implies the network is constantly “echoing” infections through highly connected nodes, resulting in continual reinfection of a constant fraction of farms. We are especially interested in studying under what conditions the infectious animal disease persists and becomes localized under regular vaccination in the intra- and interfarm model. However, besides the important role of the hub farms to stop epidemics in scale-free networks, it is also crucial to know how extensively the adverse effect is created by the low-vaccinated farms.

This work focuses on how the spatial transmission of infection is affected by both factors, the topology of networks and the internal resilience mechanism of nodes. We develop a network-based model for transmission of infectious diseases of livestock, with a focus on internal structure of each node (farm). The model characterizes two distinct dynamical regimes: intrafarm dynamics where infection spreads fast among homogeneously mixing population following conventional compartment model and interfarm dynamics where the disease transmission occurs rather slowly along farm-to-farm contact networks. Throughout this paper, we exclude culling and other control measures, since we focus on investigation of the effect of the vaccination on the transmission of FMD in the farm network. All simulation results are performed by MATLAB.

#### 2. Robustness of Networks with Resilient Nodes

Resilience of networks implies how robust a network is to accidental or intentional attack on its vertices. In many models of epidemic spreading, the close relevance of network topology in the burst of the epidemics has been confirmed [17, 22].

The susceptible-infected-susceptible (SIS) model has been studied in a network representing potential transmission of the infection. It turned out that, for a scale-free (SF) network, the threshold for epidemic spreading is null in the infinite network limit. In other words, the epidemics in a sufficiently large SF network may never be eradicated even for low spreading rates. However, this two-state model is not directly applicable to farm networks, as the nodes on such networks have more complicated internal structures. One needs to refine the inter- and intrafarm spreading process to deal with multitude of animals in the nodes at diverse infection stages.

In order to develop such hierarchical model of epidemic spreading, we first begin with a simple continuous SIS network model. The model will be elaborated with more realistic stages in the following sections. Suppose there are farms and each farm prevalence, say , , takes a continuous value between and where means absence of infection at the farm and means fully infected state; that is, all individuals in the farm are infected. Let denote an adjacency matrix whose th entry is 1 if the farms and are adjacent and the transmission occurs from the farms to , and 0 otherwise. The degree of th farm is defined as , which means the number of connected neighbors of the th farm. Then, the dynamics of the farm prevalence can be formulated as Here represents the internal epidemic development in the farm and represents the transmission rate function from the farm to the farm . If , the farm is said to be resilient from infection with the resiliency . This implies that the farm, if isolated, can recover from small infectious perturbation. In our approach, we assume that the farm-to-farm transmission of the infection is proportional to both of the prevalence at the source farm and the susceptible level at the target farm. Hence the transmission rate function is set as where is the transmission rate. We want to see under what condition farm network (1) with (2) maintains the synchronized disease-free state. The following theorem shows that a community of resilient farms is robust to infection as long as the network is not too dense and the transmission rate is low.

Theorem 1. *Consider the farm network defined by (1) and (2) that consists of resilient farms. Let be the maximum degree of the farms, and let be the minimum resiliency. Then the collective disease-free state is asymptotically stable for .*

The proof of Theorem 1 is found at Appendix. The overall resiliency of a farm network depends on the resiliency of individual farms, the network density, and the transmission rate. Figure 1 depicts the mean farm prevalence at equilibrium of thousand farms in a scale-free network according to the transmission rate . Here the simple internal process function was used to model intensive practice of vaccination which leads to exponential decay of the infection ratio. One can observe that all the farms remain disease-free where is less than a critical value around . In this parameter regime, even if some of the farms are simultaneously infected, the epidemic disease is soon eradicated from the network before it spreads further. However, such collective robustness does not hold and some of the farms fall in an endemic state if the transmission rate is greater than the critical value.