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Conceived and designed the experiments: SVS FCS JMP. Performed the experiments: SVS FCS JMP. Analyzed the data: SVS FCS JMP. Contributed reagents/materials/analysis tools: SVS FCS JMP. Wrote the paper: SVS FCS JMP.

The authors have declared that no competing interests exist.

Human societies are organized in complex webs that are constantly reshaped by a social dynamic which is influenced by the information individuals have about others. Similarly, epidemic spreading may be affected by local information that makes individuals aware of the health status of their social contacts, allowing them to avoid contact with those infected and to remain in touch with the healthy. Here we study disease dynamics in finite populations in which infection occurs along the links of a dynamical contact network whose reshaping may be biased based on each individual's health status. We adopt some of the most widely used epidemiological models, investigating the impact of the reshaping of the contact network on the disease dynamics. We derive analytical results in the limit where network reshaping occurs much faster than disease spreading and demonstrate numerically that this limit extends to a much wider range of time scales than one might anticipate. Specifically, we show that from a population-level description, disease propagation in a quickly adapting network can be formulated equivalently as disease spreading on a well-mixed population but with a rescaled infectiousness. We find that for all models studied here –

During the past decade, we learned that the structure of contact networks plays a crucial role in the spread of diseases. Most theoretical studies addressing this issue assume that contact networks are static entities, whereas the actual disease paths continuously reshape based on local social dynamics. This work aims to achieve a better understanding of disease spreading in populations characterized by a dynamically structured contact network where contacts appear and disappear over time. The network dynamics are entangled with the disease dynamics, as individuals may have access to local information that makes them aware of both the existence of the disease and the health status of their contacts, allowing them to minimize exposure to infection. Here we show the equivalence between disease propagation in an adaptive contact network and that in a well-mixed population with a rescaled transmission probability, which depends also on the fraction of infected in the population. Thus, one can emulate the effect of an adaptive contact network with a simple correction of the transmission probability. This result is obtained in the limit where network adaptation proceeds much faster than disease spreading, but we demonstrate that it also holds for a much wider range of scenarios.

During recent years it has become clear that disease spreading

As a result, quite a few studies have recently investigated the impact of dynamical networks on disease progression, as well as the influence of the way information (disease awareness) flows in parallel with disease progression and the role of noise in disease dynamics

Let us start by introducing the well established

Suppose all individuals seek to establish links at the same rate _{pq}

Often individuals prevent infection by avoiding unprotected contact with infected once they know the state of their contacts or are aware of the potential risks of such infection

The amount of information available translates into differences mostly between the break-up rates of links that may involve a potential risk for further infection (_{SI}_{IR}_{II}_{SS}_{SR}_{RR}_{I}_{H}_{I}_{H}_{SS}_{SR}_{SI}_{I}_{H}_{II}_{IR}_{RR}

In the

We would like to stress the distinction between the description of the disease dynamics at the local level and that at the population level. Strictly speaking, a dynamical network does not change the disease dynamics at the local level, meaning that infected individuals pass the disease to their neighbors with probability intrinsic to the disease itself. At the population level, on the other hand, disease progression proceeds as if the infectiousness of the disease effectively changes, as a result of the network dynamics. Hence, analyzing an adaptive network scenario at a population level can be achieved via a correction on the transmission probability, keeping the mathematically more attractive well-mixed scenario. In this sense, from a well-mixed perspective, dynamical networks contribute to change the effective infectiousness of the disease, which becomes

One can define a gradient of infection _{H}_{I}_{I}

The upper panel shows the gradient of infection _{1}

The analysis of the gradient of infection of the

Each point in the triangle (the so-called

Up to now we have assumed that the network dynamics proceeds much faster than disease spreading. This may not always be the case, and hence it is important to assess the domain of validity of this limit. In the following, we discuss the particular case of the

Circles show results of individual-based simulations for the quasi-stationary average fraction of infected

Note that the specific behavior of ^{3}. Any further increase in

Contrary to the deterministic

In the _{i}_{1}_{I}_{H}

_{I}_{I}

When recovery from the disease is impossible, a situation captured by the

Finally, in all models discussed here we also investigated the effect of allowing for different individual rates associated with the way each individual creates or destroys her social ties. Due to age-structure of most populations or intrinsic individual or cultural differences, some individuals will tend to react differently whenever they, or a contact, get infected

Making use of three standard models of epidemics involving a finite population in which infection takes place along the links of a dynamical graph, the nodes of which are occupied by individuals, we have shown analytically that the bias in graph dynamics resulting from the availability of information about the health status of others in the population induces fundamental changes in the overall dynamics of disease progression.

The network dynamics proposed here differs from those used in previous models of disease spreading on adaptive networks

The description of disease spreading as a stochastic contact process embedded in a Markov chain constitutes a second important distinction between the present model and previous studies. This approach allows for a direct comparison between analytical predictions and individual-based computer simulations, and for a detailed analysis of finite-size effects and convergence times, whose exponential growth will signal possible bistable disease scenarios. In such a framework, we were able to show that adaptive networks in which individuals may be informed about the health status of others lead to a disease whose effective infectiousness depends on the overall number of infected in the population. In other words, disease propagation on adaptive networks can be seen as mathematically equivalent to disease spreading on a well-mixed population, but with a rescaled effective infectiousness. In accord with the intuition advanced in the introduction, as long as individuals react promptly and consistently to accurate available information on whether their acquaintances are infected or not, network dynamics effectively weakens the disease burden the population suffers from. Last but not least, if disease recovery is possible, the time for disease eradication drastically reduces whenever individuals have access to accurate information about the health state of their acquaintances and use it to avoid contact with those infected. If recovery or immunity is impossible, the average time needed for a disease to spread increases significantly when such information is being used. In both cases, our model clearly shows how availability of information hinders disease progression (by means of quick action on infected, e.g., their containment via link removal), which constitutes a crucial factor to control the development of global pandemics.

Finally, it is also worth mentioning that the knowledge about the health state of the others may not always be accurate or available in time. This is for instance the case for diseases where recently infected individuals remain asymptomatic for a substantial period. The longer the incubation period associated with the disease, the less successful individuals will be in escaping infection, which reduces in our model to a lower effective rate of breaking SI links, with the above mentioned consequences. Moreover, the (social) network through which awareness of the health status of others proceeds may lead to different rates of information spread. In such cases, one may model explicitly the spread of the health state of each individual, as done in Refs.

We define disease dynamics in finite populations of size

Adopting

We obtain the finite population analogue of the well-known mean-field equations characteristic of these models by recognizing that, in the limit of large populations,

Equations (M1) and (M2) define a Markov chain

Consider a network of constant size _{pq}_{pq}

When the time scale for network update (

All individual-based simulations start from a complete network of size _{pq}^{9} disease event updates (10^{7} generations). The average number of infected ^{7} generations are shown in ^{4} independent simulations, each simulation starting with 1 infected individual. The reported results correspond to the average amount of time after which the population reaches a state with

Adaptive contact networks change effective disease infectiousness and dynamics. 1. The SIS model. 1.1 Recovery times in finite populations. 2. The SI model. 2.1. Infection times in finite populations. 2.2. Infection times in dynamical networks. 3. The SIR model. 3.1. The SIR model in finite populations. 3.2. The SIR model in dynamical networks. 4. Individual diversity in linking dynamics.

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