Computational and Mathematical Methods in Medicine

Volume 2017, Article ID 2403851, 11 pages

https://doi.org/10.1155/2017/2403851

## Stochastic Models of Emerging Infectious Disease Transmission on Adaptive Random Networks

^{1}Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand^{2}Centre of Excellence in Mathematics, CHE, Bangkok 10400, Thailand^{3}Thailand Center of Excellence in Physics, CHE, 328 Si Ayutthaya Road, Bangkok 10400, Thailand

Correspondence should be addressed to Charin Modchang; moc.liamg@gnahcdomc

Received 14 March 2017; Accepted 15 August 2017; Published 17 September 2017

Academic Editor: Delfim F. M. Torres

Copyright © 2017 Navavat Pipatsart et al. 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

We presented adaptive random network models to describe human behavioral change during epidemics and performed stochastic simulations of SIR (susceptible-infectious-recovered) epidemic models on adaptive random networks. The interplay between infectious disease dynamics and network adaptation dynamics was investigated in regard to the disease transmission and the cumulative number of infection cases. We found that the cumulative case was reduced and associated with an increasing network adaptation probability but was increased with an increasing disease transmission probability. It was found that the topological changes of the adaptive random networks were able to reduce the cumulative number of infections and also to delay the epidemic peak. Our results also suggest the existence of a critical value for the ratio of disease transmission and adaptation probabilities below which the epidemic cannot occur.

#### 1. Introduction

Within few decades, numerous studies on infectious disease transmission using the network theory have been carried out. Networks provide a mathematical platform for the interpretation of the interaction between individuals or populations and are especially useful when each individual is assumed to be in direct contact only with a small proportion of the population [1–5]. Network models tend to be very powerful tools that provide understanding of the disease transmission in human populations and allow the assumptions of either social or sexual contacts [6]. However, the vast majority of infectious disease transmission models on networks employed static networks [7]. The static network structure does not change over time and such models ignore the effect of individual’s behavioral change due to the infection. On the other hand, there are studies that implemented rules on dynamical network structures that opened the possibility of network adaptation. These rules help to generate complex network models and are also expected to reflect some real-world networks [8–13].

Recent studies brought forth characteristic rules of networks that adapted the network structure more dynamically by responding to the infection status of individuals [14–16]. These dynamic networks took into account the fact that individuals tend to respond to the emerging infectious disease transmission by avoiding contacts with infected individuals. Such rewiring of local contacts can have a strong impact on the dynamics of an infectious disease transmission as shown in a study of a complicated mutual interplay between network adaptation dynamics and the dynamics of individual states. There are two popular closely related adaptive rules of the network: the first rule allows susceptible individuals to temporarily disconnect their contacts with infectious individuals [17] and the other allows susceptible individuals to avoid contact with the infectious individuals by rewiring their network connections [18]. This has revealed new perspectives onto effects on concurrent partnerships [19, 20] and on structure changing patterns [21]. A definition of dynamic networks was given stating that such networks are regulated by a feedback loop between the dynamics of node’s state and interaction in a network and the coevolution or adaptation of the networks [22].

Most studies on infectious disease transmission using adaptive networks are deterministic. Admittedly, the implementation of such models is easier, but they are insufficient to explain some fluctuating dynamics in real-world systems [23]. Deterministic models provide exactly the same results given the same initial conditions. However, we would not expect to observe exactly the same people becoming infected at exactly the same time. In contrast, infectious disease modeling using the adaptive networks takes into account the fluctuations or noise by considering interactions between individuals. Clearly, there is an important element of chance and stochastic models are concerned with approximating this random or probabilistic element. In general, chance will play the most important role whenever the number of infectious individuals is relatively small, which can happen when the population size is small. It is especially important that stochasticity is taken into account and incorporated into the network model.

However, there is no comparative study about effect of these two adaptive rules to the infectious disease transmission in population yet. The main goals of this work is not only to model networks representing individuals in a population that tend to respond to the emergence of an infectious disease and incorporate two different specific patterns of behavioral change regarding the interaction between individuals in a population but also to investigate the interplay between infectious disease dynamics and network adaptation dynamics by using stochastic simulations in order to gain insight into infectious disease transmission in real-world system.

#### 2. Network Models

##### 2.1. Construction of Static Random Network

In our model, we represented an individual human by a node and a potential disease-causing interaction between two individuals by a link. The random network was constructed from a finite set of nodes with possible pairs. Each pair of nodes was then randomly chosen to be connected with a probability , where the number of nodes and independent probability are fixed. We defined the initial structure of random network at each simulation run by setting the initial structure as a random network. The number of individuals in this system was fixed at ,000 nodes. We expected a low frequency of the disease-causing contact between two individuals henceforth, comprising a small proportion of the population and represented by a low probability .

##### 2.2. Infectious Disease Transmission on Networks

According to the* SIR* epidemic model mechanism, an individual human was represented as a node amongst a finite and fixed number of individuals in a population and the potential disease-causing contact between two individuals as a link. Two nodes were assigned as neighbors, if connected by a link. A node neither could be linked to itself nor shares more than one link with another node. At any time, each node has only one specific state, namely, the susceptible state , the infectious state , or the recovered state . This model is appropriate for a disease that spreads through human populations by direct contact between infectious individuals and susceptible individuals, such as influenza. We also assumed that recovered individual confers lifelong immunity. In this paper we will consider only diseases of this type. Diseases that are endemic because they propagate on time scales comparable to or slower than the rate of turnover of the population or because they confer only temporary immunity are not well represented by this model.

For the → transition, an infectious disease can be transmitted, if a given infectious node shares links with other susceptible nodes. We denoted this link as -link. Those susceptible neighbors were then infected with probability per -link and per time-step. In addition, the more the contacts a susceptible node has with infectious nodes, the higher the chance to become infected simply is because a pathogen can be transmitted from many infected individuals. Hence, we calculated the infection probability of each node individually as follows:where is the number of -links that is connected to that node. The → transition implies that infected nodes can independently self-recover with a recovery probability . In contrast to the → transition, the → transition is not influenced by -links. The initial conditions were set as follow: (1) The entire nodes in the network are in susceptible state. (2) A single node is randomly chosen to become the first infectious node.

When an infectious disease spreads on a social network, humans tend to respond to the emergence of an epidemic by avoiding contacts with infected individuals. Such rewiring of the local connections can have a strong effect on the dynamics of the disease, which in turn influences the rewiring process. Thus, a complicated mutual interaction between network adaptation dynamics and the dynamics of disease transmission emerges (Figure 1). In our network, such nodes with reconnecting or rewiring ability were called adaptive nodes. It shall be noted that only susceptible nodes are capable of such adaptation once being at risk of infection. Hence, the network adaptation concerned the ability of susceptible nodes to adapt as to avoid contact with infectious nodes. In addition, these behavioral changes can lead to the isolation of infected nodes and then again affects the behavioral changes of susceptible nodes. Consequently, there is interplay between the changing state of nodes and the changing interaction of nodes at any time of infectious disease transmission on the network. This provides a feedback loop between the infectious disease dynamics and network adaptation dynamics of the network as illustrated in Figure 1. These networks including feedback loops are defined as coevolution networks or adaptive networks [24].