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Complex dynamics of synergistic coinfections on realistically clustered networks

Edited by Simon A. Levin, Princeton University, Princeton, NJ, and approved June 25, 2015 (received for review April 21, 2015)
Significance
Concurrent infection with multiple pathogens is an important factor for human disease. For example, rates of Streptococcus pneumoniae carriage (a leading cause of pneumonia) in children under five years can exceed 80%, and coinfection with other respiratory infections (e.g., influenza) can increase mortality drastically; despite this, examination of interacting coinfections on realistic human contact structures remains an understudied problem in epidemiology and network science. Here we show that clustering of contacts, which usually hinders disease spread, can speed up spread of both diseases by keeping synergistic infections together and that a microscopic change in transmission rates can cause a macroscopic change in expected epidemic size, such that clustered networks can sustain diseases that would otherwise die in random networks.
Abstract
We investigate the impact of contact structure clustering on the dynamics of multiple diseases interacting through coinfection of a single individual, two problems typically studied independently. We highlight how clustering, which is well known to hinder propagation of diseases, can actually speed up epidemic propagation in the context of synergistic coinfections if the strength of the coupling matches that of the clustering. We also show that such dynamics lead to a firstorder transition in endemic states, where small changes in transmissibility of the diseases can lead to explosive outbreaks and regions where these explosive outbreaks can only happen on clustered networks. We develop a meanfield model of coinfection of two diseases following susceptibleinfectioussusceptible dynamics, which is allowed to interact on a general class of modular networks. We also introduce a criterion based on tertiary infections that yields precise analytical estimates of when clustering will lead to faster propagation than nonclustered networks. Our results carry importance for epidemiology, mathematical modeling, and the propagation of interacting phenomena in general. We make a call for more detailed epidemiological data of interacting coinfections.
Individuals are at constant attack from infectious pathogens. Coinfection with two or more pathogens is common and can seriously alter the course of each infection from its own natural history. Infection with HIV increases susceptibility to many pathogens, especially tuberculosis, where coinfection worsens outcomes and increases transmission of both pathogens (1). Recent studies have examined epidemiological case counts to highlight the importance of upper respiratory infections (e.g., rhinovirus, influenza virus, respiratory syncytial virus [RSV]) and Streptococcus pneumoniae carriage leading to increased risk of pneumococcal pneumonia (2 ⇓ ⇓ –5), although there are few dynamic transmission models of pneumococcus (PC) and other viral infections.
Models of disease transmission in structured populations have remained a main focus of network theory for over a decade as realistic descriptions of contact structures are necessary to understand how diseases are transmitted between individuals (6 ⇓ ⇓ ⇓ ⇓ –11). Typically, specific structural properties (average degree, network size, clustering) are explored in isolation. It remains a strong (and potentially dangerous) assumption that results obtained with different models exploring distinct structural properties will give the same results when combined with other models exploring different properties. Disease transmission is a nonlinear problem with features of the propagation itself interacting in complex ways. In this paper, we focus on combining two much studied phenomena—realistic clustering of contact structure and the interaction of respiratory pathogens (influenza and PC pneumonia)—and show that a combination of these two phenomena leads to behavior that is unexpected given previous studies.
An impressive amount of research has focused on the impact of clustering on disease dynamics (12 ⇓ ⇓ ⇓ ⇓ ⇓ ⇓ –19). Clustering is often simply described as the number of triangles (where the friend of my friend is also my friend) in a network, but usually also implies that links between nodes tend to be aggregated in wellconnected groups. This aggregation tends to hinder the spread of the disease by keeping it within groups where links are more likely to connect to already infected (immune) nodes (18). Clustering plays an important role in Ebola virus transmission (20), respiratory infections (21, 22), and sexually transmitted infections (23, 24).
On the other hand, the interaction of two spreading agents has received a great amount of attention mostly due to the generality of such models (25, 26). These spreading agents can represent two different, but interacting, diseases (such as sexually transmitted infections) (26, 27); the propagation of awareness campaigns trying to stop the spread of an epidemic (25); or even the competition between a mutated strain of influenza and the original strain (10, 11, 28). These dynamics can by themselves exhibit complex behaviors; however, we will see here that they can be further influenced in equally complex ways by the structure imposed on the underlying network.
Here we describe a susceptibleinfectioussusceptible (SIS) network model incorporating variable clustering strength and two interacting pathogens and provide a meanfield formalism to follow its dynamics. We find that synergistic coinfections can lead to faster disease spread on clustered networks than on an equivalent random network, contrary to previous studies considering single infections (18). We introduce a criterion based on tertiary infections (or twostep branching factor), which allows analytical prediction of whether a clustered or random network propagates infections most efficiently. Finally, we observe a firstorder phase transition in epidemic final size, meaning that a microscopic change in transmissibility can lead to a macroscopic (and discontinuous) increase in disease prevalence. We also identify a dangerous parameter region where an infection would propagate in a clustered network but not in a random network, indicating that movement of coinfected individuals into new clustered networks of susceptible individuals could cause explosive outbreaks.
Network Structure and Epidemic Dynamics
We extend a recent description of propagation dynamics on a highly clustered network using overlapping community structure (18). This particular arrangement of nodes leads to the aggregation (or clustering) of nodes into wellconnected groups, representing for example a person’s family or workplace. Every connection in this structure can be decomposed in terms of groups, where even single links between two individuals can be considered as a group of size two. Assuming that we know the distribution of group sizes (number of nodes per group) and of node memberships (number of groups per node), we can define a maximally random ensemble of clustered networks with a fixed community structure by randomly assigning nodes to groups (Fig. 1). Hence, the entire network structure is solely defined by two probability distributions,
The dynamics of a single disease on this community structure model was studied in ref. 18. Using a meanfield description, it was shown how the clustering of links in groups slowed down propagation as links are wasted on redundant (immune) connections instead of reaching new (susceptible) individuals. Expanding this study, we now study the effects of having two diseases interacting on these clustered networks.
To highlight the effects of community structured (CS) vs. random networks (RNs), both topologies will be studied analytically and numerically. The CS network will be compared with its equivalent random network (ERN): a network with exactly the same degree distribution but with randomly rewired links. This rewiring is done by setting
On these networks, we study the dynamics of two diseases exhibiting SIS dynamics. Without interaction with the other disease, an infectious individual would infect its susceptible neighbors with disease i at rate
Similarly, we will distinguish groups by their size n and the states of the nodes they contain. Thus,
A meanfield description of the time evolution can be written in the spirit of existing formalisms (18, 29). Here we give a brief description of how the meanfield equations are obtained and provide the full system in
SI Appendix
. The meanfield description of the dynamics follows the rates that individuals move from one state to another. For instance, the fraction of individuals infected by disease 1 and susceptible to disease 2 will change as
The evolution of group states can be followed by a single, albeit more complicated, equation. This equation governs the rate of change in
Combining all possible recovery and infection terms yields the full equations as provided in
SI Appendix
. This new set of equations is coupled to the previous one through the meanfield of excess interactions
Finally, to close the model, we simply write the basic interaction meanfields (average interaction within a given group) with the available information. For example, we find for disease 1
Intuitively, for
Results
We validate our meanfield formalism with simulations of coinfections on highly clustered networks. We will focus on two diseases that can interact synergistically with their respective propagation, i.e., with
Fig. 2A presents the effects of clustering with noninteracting diseases and the same scenario in the presence of synergistically interacting diseases. We find that although clustering slows down the propagation of noninteracting diseases, it speeds up the propagation of synergistically interacting diseases. This result is important for network models of disease: random networks are considered worst case scenarios for the speed of disease propagation (18, 21), implying that models can justify working in a random network paradigm. However, this is clearly not always the case in the presence of interacting diseases with synergistic effect.
Clustering Threshold.
We are now interested in identifying a simple analytical criterion, as a function of the clustering coefficient C, and the disease interaction parameters α, to determine when a clustered network structure is more efficient at synergistic disease transmission than a random structure. Because we are interested in the relative speed at which a pair of diseases move through a population, we can generalize the idea of the basic reproductive number,
As the calculation of
The effect of this clustering coefficient is twofold: first, further infections are conditional on links not wasted with infected neighbors of the root node, and second, in the event of a single infection (probability
This approach is validated on Fig. 2B . Once again, we use networks constructed from cliques of size 10, such that clustering not only comes through the root node but also from other newly infected nodes. We also examine another clustered network, with the same degree distribution, but composed only of triangles, leading to a clustering coefficient around 0.06. We can see that our approach is able to give precise estimates of the coupling strength for which both clustered networks start spreading faster than their equivalent random networks.
Fig. 3 demonstrates this over a broad range of parameters. For realistic ranges of clustering and disease interaction, we find faster propagation on clustered networks than random. Of course, for very clustered networks (i.e., when
FirstOrder Transitions and Outbreak Risk.
We have thus far investigated the impacts of varying the coupling α between the two diseases. However, impact of clustering is expected to be stronger for more transmissible diseases (larger
This standard assumption is not always justified, however. Fig. 4 presents endemic steady states for clustered and random networks over a range of transmission rates
Coupling these two surprising results, we see that for a critical range of parameters, a microscopic increase in transmissibility can cause a macroscopic difference in the expected epidemic size on a clustered network but not on an otherwise equivalent random network. This conclusion is confirmed in the shaded region of Fig. 4A
: the diseases spread to around
In the context of diseases that spread heavily in daycares and schools, this means that a small difference in the clustering of contacts could translate to a difference between no outbreak and a complete contagion for interacting pathogens such as influenza and PC pneumonia.
Discussion
Here we demonstrated that synergistic coinfections, such as pneumonia caused by S. pneumoniae and influenza, may actually spread faster and farther on clustered networks than on random networks. This result is similar to the recent observation that behaviors or opinions can propagate more rapidly in clustered social networks than in their random equivalent due to social reinforcement (34, 35). Our model thus suggests that we could also expect to see faster transmission on clustered networks in the context of diseases requiring multiple exposures before infection, which can also lead to discontinuous phase transitions (33).
We identified a threshold above which a clustered network structure will enhance the spread of synergistic coinfections. Finally, we demonstrated a firstorder phase transition in final epidemic size and identified regions where coinfected individuals can start large outbreaks on clustered networks where they wouldn’t on random networks. Our results provided here have clear implications for understanding transmission dynamics of interacting diseases on realistic contact networks and for network based modeling of infectious disease transmission.
Understanding how diseases interact within host and between hosts in populations with realistic contact structure is of key importance to epidemiologists working to limit the transmission of diseases in these populations. Pneumococcal carriage rates in children under five years old interacting in highly clustered communities (daycares and households) can exceed 80% (36), and influenza infections are common. According to our results, transplanting a child coinfected with PC and influenza into a new clustered setting with susceptible hosts could result in an unexpectedly large outbreak. Similar outbreaks of sexually transmitted diseases could occur if individuals coinfected with syphilis and HIV (27), for example, entered into a clustered network of susceptible individuals, such as prostitution networks (24).
Our results are of importance to the field of epidemic modeling in general. Common practice is to run epidemic dynamics on random networks or mass action models, as these are considered worst case scenarios for transmission. We showed that network clustering facilitates synergistically interacting diseases because tight clustering keeps the diseases together. Our clustering threshold can be used by modelers to test whether they should be considering random or clustered network dynamics when trying to identify pessimistic transmission scenarios.
Our study has implications for epidemiology, mathematical modeling, and for the understanding the propagation of interacting phenomena in general. However, as illustrated by the problems encountered in trying to identify ranges of realistic parameters (Fig. 3), there is a dire need for data in the context of interacting epidemics. Not only is it hard to estimate realistic contact network properties, but one would also need to be able to estimate the transmissibility of a pathogen in both the absence and presence of other possible interacting diseases. Therefore, although this work is a step forward in terms of theory, it should also be taken as a call for better data.
Acknowledgments
This work was supported by the Santa Fe Institute, the James S. McDonnell Foundation Postdoctoral fellowship (L.H.D.), and the Santa Fe Institute Omidyar Postdoctoral fellowship (B.M.A.).
Footnotes
 ↵ ^{1}To whom correspondence should be addressed. Email: laurent{at}santafe.edu.

Author contributions: L.H.D. and B.M.A. designed research, performed research, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1507820112//DCSupplemental.
Freely available online through the PNAS open access option.
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