We study the spreading of an infection within an SIS epidemiological model on a network. Susceptible agents are given the opportunity of breaking their links with infected agents. Broken links are either permanently removed or reconnected with the rest of the population. Thus, the network coevolves with the population as the infection progresses. We show that a moderate reconnection frequency is enough to completely suppress the infection. A partial, rather weak isolation of infected agents suffices to eliminate the endemic state.
Outbursts of epidemics in human populations trigger individual and collective reactions that can substantially alter the social structure. As a consequence of risk perception, non-infected individuals may start avoiding contact with their infected equals, even when their previous relationship was fluent. The whole society could collectively decide to isolate its infected members until danger is overcome. More altruistic non-infected individuals may be tolerant of the contact with infected individuals but, in turn, the latter may discontinue the relationship to impede contagion. The escape from crowded cities during the Black Death in the late Middle Ages, documented in Giovanni Boccaccio's Decameron, and the closing of schools, churches, and theaters during influenza epidemics in the early twentieth century, constitute dramatic historical instances of such behaviours [1,2]. Quarantine protocols, and preventive isolation during leprosy or tuberculosis treatment, are present-day examples [3]. In any case, these changes in the pattern of social contacts help to limit and control the incidence of the infection.
In this paper, we explore the effects of an evolving pattern of contacts on the dynamics of infection spreading, in the framework of a simple epidemiological model. We consider a population of agents whose pattern of contacts is represented by a network. If a link of the network joins two agents, contagion is possible when one of them is infected and his neighbour is not. To account for the social processes addressed in the previous paragraph, we admit that the contact network is not a static structure, but evolves in response to the epidemiological state of the population.
Agent-based models whose interaction patterns are represented by networks have received increasing attention during the last years, in the analysis of emergent collective behaviour in complex systems [4]. Frequently, the evolution of the interaction network and the dynamics of individual agents occur over different time scales. In learning processes, for instance, connections change adaptively over scales that are large as compared with the internal dynamics of agents [5]. At the opposite limit, in models of network growth, the pattern evolves in the absence of any dynamics related to the agents [4]. When, on the other hand, the dynamical time scales of a population of agents and its interaction network are comparable, we can speak about their coevolution [6,7,8,9,10]. In this context, the model considered in the present study can be regarded as an illustration of the coevolution of agents and networks, inspired in the dynamics of infection spreading.
Our model is based on an SIS epidemiological process where, at a given time, each agent can be susceptible (S) or infected (I). In the standard SIS process, each I-agent spontaneously recovers and becomes susceptible at a fixed rate, say, with probability γ per unit time. An S-agent, in turn, becomes infected by contagion from his infected neighbours. If the contagion probability per unit time and per infected neighbour is ρ, an S-agent with k I infected neighbours becomes itself infected with probability
during the interval dt. Within a mean-field description, if the average number of (both S and I) neighbours per agent is k and the fraction of I-agents is n I , we have k I = kn I . The mean-field evolution equation for n I thus reads ṅI = -γn
where n S = 1 -n I is the fraction of S-agents, and λ = kρ. In this description, for asymptotically long times, n I vanishes if λ ≤ γ. Therefore, the infection is suppressed as time elapses. If, on the other hand, λ > γ, the fraction of infected agents approaches a finite value n * I = 1 -γ/λ > 0, and the infection is endemic. The transition between these two regimes occurs through a transcritical bifurcation at λ = γ.
In the following, we complement the standard SIS model with the possibility that the network of contacts changes in response to the infection spreading. Specifically, links between susceptible and infected agents can be broken, and either removed or reconnected to other agents. As expected, we find that this mechanism decreases the infection level, and can eventually suppress the endemic state. With respect to the standard model, however, infection suppression for high rates of contact change occurs through a tangent bifurcation, which in turn gives rise to a bistability regime. In this regime, the infection persists or dies out depending on the initial fraction of infected agents. More unexpectedly, infection suppression does not require a drastic overall change in the network structure, but is reached with a moderate unbalance between the mean number of neighbours of susceptible and infected agents. As we discuss in the final section, these features are robust under several variations of the dynamical rules.
We address first the case where before the interaction between an S-agent and an I-agent joined by a network link eff
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