Epidemic forecast is crucially depending on our ability to model the spread of epidemics in spatially extended systems and the movement of individuals at various levels, from the global scale of transportation flows to the local scale of the activities and contacts of individuals. In this context, modelling in mathematical and statistical epidemiology has evolved from simple compartmental models into structured approaches in which the heterogeneities and details of the population and system under study are becoming increasingly important features [1] (see Fig. 1). In the case of spatially extended systems, modelling approaches have been extended into schemes that explicitly include spatial structures and consist of multiple sub-populations coupled by travelling fluxes, while the epidemic within the sub-population is described according to approximations depending on the specific case studied [2–10]. This patch or meta-population modelling framework has then grown into a multiscale framework in which the various possible granularities of the system (country, intercity, intracity) are considered through different approximations and coupled through interaction networks describing the flows of people and/or animals [10–17]. At the most detailed level, the introduction of agent-based models (ABM) has enabled to stretch even more the usual modelling perspective, by simulating the propagation of an infectious disease individual by individual [18,19].

The above modelling approaches are based on actual and detailed data on the activity of individuals, their interactions and movement, as well as the spatial structure of the environment, transportation infrastructures, traffic networks, and travel times. While for a long time this kind of data was limited and scant, recent years have witnessed a tremendous progress in data gathering, thanks to the development of new data-processing tools and the increase in computational power. A huge amount of data, collected and meticulously catalogued, has become finally available for scientific analysis and study. The scientific community has subsequently uncovered in such data the presence of complex properties and heterogeneities that cannot be neglected in epidemic-modelling description. In particular, the ever-increasing level of interconnectedness and globalization of our modern society along with a high level of diversity and heterogeneity induces a novel epidemiological context: the mathematical and computational modelling of disease spread needs to integrate such complex features.

Although there is no commonly accepted definition of complex systems, they share a number of characteristics. They are made of a large number of interacting components and there is not a global blueprint controlling their evolution. One of the most peculiar features of complex systems is their non-trivial collective behaviour and their resilience to perturbations; i.e. their ability to adapt to a fluctuating environment and to evolve. This last point in particular allows one to distinguish complex from complicated systems, since random perturbations on complicated systems will lead in most cases to their failure.

These considerations might appear as a call for a modelling approach of complex systems that considers as many possible parameters and details as we can possibly handle. The debate about the amount of “realism, precision, and generality” needed in a model is not new, but still very vivid [20,21]. As noted by May and previous authors (see [20] and references therein), there is a broad spectrum of models ranging from ‘toy’ models to highly detailed models. Toy models sacrifice precision and sometimes realism in order to capture the essence of the phenomenon and the general mechanisms. On the other side of the spectrum, models with a high level of detail provide the opportunity of analyzing the spreading process in a very realistic way, making all assumptions explicit, the main drawback being that the key mechanisms underlying the epidemic evolution are difficult to identify and discriminate because of the numerous assumptions and of the large number of elements of the system. It is thus a difficult task to obtain finally a model that stays at a reasonable level of precision, but still captures enough realism to be useful in practical situations, such as forecasting and control strategies' assessment. In the case of complex systems, this task is simplified by the fact that it is possible to distinguish different classes of parameters and to identify which ones are really relevant in the description of the large-scale behaviour of the system. By leveraging on the recent understanding of complex systems, it is then possible to complement the basic approaches developed in mathematical and statistical epidemiology with the introduction of large scale systems (10⁴ to 10⁶ degrees of freedom), and to isolate the main features responsible for their behaviour. This allows the systematic investigation of the impact of the various complex features of real systems on the basic properties of epidemic spreading.

In the following, we provide a discussion of the various instances at which the inclusion of complex features is relevant in epidemic modelling. We will then discuss the spread of epidemics in complex networks and the effect of large-degree fluctuations on the spreading process. In Section 4, we discuss a further step towards realism with the implementation of metapopulation models. In this Section, we will focus especially on the new directions emerged in the last years in the context of large computational approaches that pave the way to the establishment of computational infrastructures able to provide quantitative forecast of epidemic spreading.

In the pre-industrial times, disease spread was mainly a spatial diffusion phenomenon. For instance, during the spread of the so-called Black Death, which occurred in the 14th century, only few travelling means were available and typical trips were limited to relatively short distances on the time scale of one day.

Historical studies confirm that the propagation (Fig. 2) indeed followed a simple scheme, with a spatio-temporal spread mainly dominated by spatial diffusion. More precisely, ballpark calculations on the historical data show that the Black Death essentially spread through Europe from south to north, with the invasion front moving at an approximate velocity of 200–400 miles/year [22]. Mathematically, this process can be described with a simple Susceptible-Infected-Removed (SIR) model with diffusion, which can be written as:

Our modern societies are sharply contrasting with the previous example: due to the large variety of travelling means with different distance and time-scales associated, epidemiology cannot simply rely on an approximate spatial description of the disease spread. The interplay between social networks and infrastructures was argued to be at the origin of the spreading pattern of a disease already observed in the 19th century, when the English physician John Snow analyzed on a map the relation between the public water-supply system and cholera cases in London [23]. As anticipated moreover already in 1933 [24], the large scale and geographical impact of infectious diseases on populations in the modern world is mainly due to commercial air travel. This has been repeatedly and dramatically demonstrated in several circumstances, such as the international airline hub-to-hub pandemic spread of acute hemorrhagic conjunctivitis in 1981 [25], and more recently the evolution of SARS epidemic [26]. While this epidemic indeed first diffused out of its origin in China and spread in South-East Asia, it also reached very rapidly much farther regions, such as North America and Europe, brought by infected individuals travelling by plane (see Fig. 3). One of the most dangerous aspects of the SARS epidemic was in fact its very fast spread on worldwide scales. This picture, therefore, cannot be simply described in terms of diffusive phenomena, but ought to explicitly incorporate the spatial structure of modern transportation networks [27,28], which have been identified as one of the main mechanisms for propagation on a global scale. In this perspective, epidemic modelling thus changes from a description in terms of local diffusion to one in which long-range interactions (i.e. flights connecting far apart airports) play a crucial role. The identification and characterization of the underlying network of infrastructures is therefore fundamental.

Networks that trace the activities and interactions of individuals, social patterns, transportation fluxes, and population movements on a local and global scale [18,27,29,30] are therefore the key ingredients in our understanding of epidemic behaviour and in our capabilities to predict their evolution. The importance of networks in epidemiology has not to be stressed here. The population connectivity pattern and the structure of meta-population models are for example naturally defined in terms of networks. The recent progresses in the field stem from the increased ability to gather data on several large sets of networked structures and populations, finding that they exhibit complex features encoded in large-scale heterogeneity, self-organization and other properties typical of complex systems [31–33]. While heterogeneity has been acknowledged since long as a relevant factor in determining the properties of epidemic spreading phenomena [1,34], many real world networks exhibit levels of heterogeneity that were not anticipated until a few years ago. Moreover, the presence of such features was recently found to have a strong impact on the resulting infection dynamics, breaking down the standard epidemiological framework. In order to discuss the relevance of complex systems' properties on the spread of an infectious disease, we will distinguish in the following between two different levels of modelling approaches corresponding to two different granularity scales – the population and meta-population description levels.

2.1 Single population level

2.2 Meta-population level

The heterogeneity found in the connectivity pattern of the contact networks described in the previous section has a strong impact on the properties of the dynamical processes occurring on the networks. A striking example is provided by the different expressions of the necessary condition for the spread of a disease across a population, depending on the homogeneous or heterogeneous character of the contact network of this population.

In the framework of homogeneous approximations, the SIR model described by Eq. (1) leads to a major outbreak, thus infecting a finite fraction of the population, if the following condition – called epidemic threshold – is satisfied: S0β/μ>1 [1,22,50]. This condition simply states that in order to have a non-zero fraction of the population infected by the virus, the rate at which infectious individuals are generated must be larger than the rate at which they recover from the disease.

In order to account for contact networks in which different individuals can have very different numbers of contacts (degrees), Eq. (1) has to be generalized to describe the evolution of the numbers of infected individuals of degree k. It can then be shown that the disease will affect a finite fraction of the population if βS0μ>〈k〉〈k2〉 [39,40]. For homogeneous patterns of contacts, 〈k2〉 is finite, but the picture is completely different when the underlying network on which the infection dynamics occurs displays large-degree fluctuations, i.e. when the degree distribution is heavy-tailed. In particular, scale-free networks with degree exponent 2<γ⩽3, for which 〈k2〉→∞ in the limit of an infinitely large network, experience a null epidemic threshold, since the above condition is always satisfied. Even diseases with very low transmission probabilities are thus able to cause a major outbreak infecting a finite fraction of the nodes. This new scenario is of practical interest in computer virus diffusion and for the spreading of diseases in heterogeneous populations [29,30,32,33,40]. It also raises new questions on how to protect the network and find optimal strategies for the deployment of immunization resources [41,42].

The previous result on the epidemic threshold concerns the stationary properties of endemic states or the final percentage of cases of an epidemic. The impact of topological heterogeneities of the contact network on the dynamical evolution of the outbreaks has also been investigated [43]. It turns out that the time behaviour of epidemic outbreaks and the growth of the number of infected individuals are governed by a time scale τ proportional to the ratio between the first and second moment of the network's degree distribution:

The relevant impact of heterogeneity and connectivity fluctuations in the epidemic modelling at the level of a single population leads us to investigate the role of complex connectivity patterns and traffic in meta-population models. Let us consider as a prototypical case the worldwide spreading of epidemics through air travel. As a basic modelling strategy, it is possible to use a meta-population approach [7–9] in which individuals are allowed to travel from one city to another by means of the airline transportation network, while the disease within the city is described with opportune compartmental models or more detailed description of the disease dynamics. This amounts to write for each urban area the set of equations:

This modelling program dates back to the work of Rvachev and Longini [11], and it has been used along the years to simulate diseases such as pandemic influenza [44–46], HIV [47], and SARS [13]. While these earlier studies were considering a limited number of urban areas and travel connections, it has recently become possible to scale up this approach by including the full International Air Transport Association (IATA) [48] database. This has led to a modelling framework [16,17] considering up to 3100 airports with demographic data for the surrounding urban areas and 17 182 connections among them, each representing the presence of a direct flight. This corresponds to more than 99% of the worldwide commercial traffic by plane. To each link connecting airports i and j is attached the weight wij, given by the number of passengers travelling on that route in a given time (e.g., on a daily basis). The inclusion of such an extensive database is motivated by the various levels of complexity and heterogeneity present in the system composed by the worldwide air transportation network (WAN) and the associated urban areas. In particular, it is possible to identify three relevant levels of strong fluctuations: the topology of the airport network, the distributions of the numbers of passengers and of the city populations (see Fig. 4).

The model obtained by integrating all these data and the aetiology of the disease within each city can be used to forecast the behaviour of emerging diseases as well as to validate the approach. Strikingly, this modelling appears to provide very good results in agreement with historical data [13,49], thus spurring the issue of identifying the fundamental limits in epidemic evolution predictability with computational modelling and their dependence on the underlying complex features of the system.

A major question in the modelling of global epidemics consists indeed in providing adequate information on the reliability of the obtained epidemic forecast, i.e. the epidemic predictability. The intrinsic stochasticity of the epidemic spreading will make each realization unique and reasonable forecast can be obtained only if all epidemic outbreak realizations starting with the same initial conditions and subject to different noise realizations are reasonably similar. A convenient quantity to monitor in this respect is the vector π→(t), whose components are πj(t)=Ij(t)/∑Il; i.e. the normalized probability that an infected individual is in city j. The similarity between two outbreak realizations (I and II) is quantitatively measured by the statistical similarity of the vectors π→I(t) and π→II(t). Such a measure of similarity sim(π→I,π→II) is given by the standard Hellinger affinity: sim(π→I,π→II)=∑πjIπjII. Possible differences in the total (worldwide) epidemic prevalence i=∑Ij/P (where P is the worldwide population) are moreover measured by sim(i→I,i→II) where i→I(II)=(iI(II),1−iI(II)). The overlap function measuring the similarity between two different outbreak realizations is thus defined by:

If we consider a model in which the cities are linked by a completely homogeneous transport network (HOMN), where both degrees of each city and traffic flows on each connection are close to their average values, we find a significant overlap (Θ(t)>80%, see Fig. 5) even at the early stage of the epidemics – the most relevant phase for epidemic surveillance and the more prone to stochastic fluctuations. The picture is different if we consider a heterogeneous topology (with both badly connected airports and hubs) associated to homogeneous travel fluxes (HETN), since especially at the initial stage of the epidemics the predictability is much smaller. Finally, the values of the overlap for epidemics propagating on the real air-transportation network (WAN) show an intermediate situation.

These results may be rationalized by considering the conflicting effects of the various levels of heterogeneity (see Fig. 6). On the one hand, the heterogeneity of the connectivity pattern (broad distribution of degrees), and in particular the existence of hubs, provides a multiplicity of equivalent channels for the travel of infected individuals, depressing the predictability of the evolution, as the comparison of HETN and HOMN shows. On the other hand, the heterogeneity of traffic flows introduces dominant connections that select preferential pathways, increasing the epidemic predictability. The backbone of such dominant spreading channels thus defines specific ‘epidemic pathways’, which are weakly affected by the stochastic noise. In the case of the worldwide airport network, the heterogeneity of the fluxes thus partially compensates for the decrease in predictability due to the topological heterogeneity.

In summary, different levels of complexity of the system affect our ability to predict the spatio-temporal spread of a disease in opposite ways. The intrinsic stochastic nature of the propagation of directly transmitted diseases – inherent both in the infection dynamics and in the movements of individuals – makes it harder to forecast the process evolution on a complex pattern of interactions. However, an additional level of heterogeneity, here encoded in the broad distribution of passenger travel flows, plays a crucial role at our advantage, making epidemic forecasts a feasible problem to address. This result clearly shows how the interplay and the integration of several levels of complexity of the system produce unexpected phenomena, which needs to be accounted for in order to obtain a better general understanding of the process.

Taking into account the complexity of real systems in epidemic modelling has shown to be unavoidable, and the corresponding approaches have already produced a wealth of interesting results. While this has stimulated the recent focus on large-scale computational approach to epidemic modelling, it is clear that many basic theoretical questions are still open. How does the complex nature of the real world affect our predictive capabilities in the realm of computational epidemiology? What are the fundamental limits in epidemic evolution predictability with computational modelling? How do they depend on the level of accuracy of our description and knowledge on the state of the system? Tackling such questions necessitates exploiting several techniques and approaches. Complex systems and networks analysis, mathematical biology, statistics, non-equilibrium statistical physics, and computer science are all playing an important role in the development of a modern computational epidemiology approach. While such an integrated approach might still be in its first steps, it seems now possible to imagine ambitiously the creation of computational epidemic forecast infrastructures able to provide reliable, detailed, and quantitatively accurate predictions of global epidemic spread.