Tag Archives: simulation

The Little Mouse on the Prairie

We have a new paper in the Early Edition of PNAS on the ecology of plague in prairie dogs. The Stanford News Service did a nice little write-up of the paper (and Mark Shwartz’s full version is available on the Woods Institute site) and it has now been picked up by a number of media outlets including USA Today, ScienceDaily, The Register (UK), as well as a couple of radio news shows. This paper has been a real pleasure for me because of my incredible collaborators.  Dan Salkeld, who has been a post-doctoral fellow with me and now splits his time between teaching in Human Biology at Stanford and working as an epidemiologist for the California Department of Health, is the lead author.  Dan is clearly one of the leading young disease ecologists working today and his understanding of the field and willingness to do the sometimes unglamorous grunt work of ecology in pursuit of important research questions continually impresses me. The paper uses data that he collected while he worked for co-author Paul Stapp on Paul and collaborators’ plague project in the Pawnee National Grasslands in Colorado. Dan and Paul had the idea that grasshopper mice (see below) might have something to do with the episodic plague outbreaks in prairie dog towns.  Apparently, this idea was met with skepticism by their colleagues. When Dan came to Stanford, I suggested that we could probably put together a model to test the hypothesis. While we were waiting for our research permits to come through for a project in Indonesia (also dealing with plague; another long story), we decided to take up the challenge. What really made the whole project come together was the fortuitous office-pairing of Dan with Marcel Salathé, another post-doc with whom I have collaborated extensively on questions of social networks and infectious disease.  In addition to being a brilliant theoretical biologist, Marcel is an ace Java programmer.  Following a few white-board sessions in the studio near our offices, Dan and Marcel put together an amazing computer simulation that achieves that perfect balance between simplicity and realism that allows for scientific insight.

I don’t think anyone would have predicted this particular collaboration and this particular outcome.  The results described in this paper come from an incredibly interdisciplinary collaboration. I am really struck at how great science can come from a few simple ingredients: (1) long-term ecological data collection facilitated by a visionary program at the National Science Foundation, (2) a space where people from quite different disciplines and with different scientific sensibilities can get together and brain-storm, (3) flexible funding that permits researchers to explore the interesting – if offbeat – scientific questions that arise from such interactions.  So, I have many debts to acknowledge for this one.  The field data come from the project for which Mike Antolin at Colorado State is the PI (out co-author Paul Stapp is a Co-PI for that as well).  The funding source for this project was the joint NSF/NIH Ecology of Infectious Disease program. This is a cross-cutting program that “supports the development of predictive models and the discovery of principles governing the transmission dynamics of infectious disease agents” (from the EID home page).  The space – both physical and intellectual – that permitted this work to happen was provided by the Woods Institute for the Environment.  This paper literally came into being in the project studio on the third floor of Y2E2 in the Land Use and Conservation area.  Amazingly, this is exactly what these studios were designed to do.  My office in Y2E2 has adjoining office space for grad-students and post-docs and this is where Marcel and Dan did most of their hashing. It was always amusing to pop my head in and see them both huddled around a computer, having animated discussions about how best to represent the complex ecology in a computational model that is simple enough to understand and flexible enough to allow us to test hypotheses.  Finally, funding. Dan was funded by a Woods Environmental Ventures Project grant for which I am the PI.  Marcel was funded by the Branco Weiss Science in Society Fellowship.  My own flexibility was assured by a career grant form the National Institutes of Health. Research funding is almost always important, but the requirements of research funding can sometimes be too constraining to permit exploration of really new ideas.  All three of these mechanisms (Woods EVP, Branco Weiss, NIH K01) provide exactly the type of flexibility that fosters creativity. I wish there were more programs like these.

One of the fundamental questions in disease ecology is how extremely pathogenic infectious agents persist both through time and across landscapes.  Plague is a bacterial disease that affects a wide range of rodents throughout the world and, in North America, particularly afflicts prairie dogs (Cynomys ludovicianus).  Plague epizootics (the animal equivalent of epidemics in humans) are dramatic affairs with almost complete mortality of massive prairie dog ‘towns’ of thousands of animals. If plague is so deadly to prairie dogs, how does it persist?  Is there another reservoir (i.e., an other host species that can maintain an infection in the absence of prairie dogs)?  Does plague get into the soil and persist in some sort of suspended state (the way that some Mycobacteria do, for example) waiting to reinfect a re-colonized prairie dog town? Or is plague really enzootic (i.e., when an infection persists at low levels in an animal population) and we just haven’t detected it? This question has wide applicability. Consider diseases of people such as Ebola Hemorrhagic Fever or SARS, or, going back a few hundred years in human history, that nastiest of bacterial diseases, bubonic plague. Yes, the same beastie.  A disease that killed a third of the population of Europe in the fourteenth century exists in prairie dogs in North America today (and sometimes spills over to produce human infections).

Prairie dogs are a keystone species of the grasslands of the American West. They are  threatened by various anthropogenic forces, including habitat destruction and human persecution.  But most importantly, prairie dog viability is threatened by plague.

Plague, a disease caused by the bacterium (Yersinia pestis) and the causative agent of Black Death, arrived in USA via San Francisco ca. 1900, and still infects (or threatens to infect) people each year, including in California. Plague killed as many as 200 million people in Medieval Europe. It is still important in Africa and Asia.  There have been sizable epidemics as recently as the middle twentieth century in India and China and a substantial outbreak in Surat, India in 1994 that, in addition to death, caused widespread panic and social disruption.

Previous modeling and ecological work tended to assume that die-outs occur very rapidly.  But questions dogged this work (as it were): were the apparently rapid die-offs simply an artifact of finally seeing dead dogs dropping all over the place? Prairie dogs do live underground, after all, and they live in enormous towns.  Who would miss a few dead dogs underground in a town of thousands?  Our paper suggests that previous modeling efforts get the story wrong. They fail to account for observed patterns because they missed key elements of the picture.  Previous models that could describe the phenomena lacked an actual explanation – it’s a magical reservoir? It’s a carnivore? Certainly it’s something somewhere?

While prairie dogs live in enormous towns, they are highly territorial within the towns.  Towns form because of the benefits of predator defense. They live in small family groups known as coteries, and these coteries form a more-or-less regular grid of small defended territories within the towns. Because of this regular structure induced by their territoriality, a directly-transmitted infectious disease can only move so quickly through a town since it could only be transmitted to immediate neighbors and each coterie only has a couple of these.  Plague is not directly transmitted though.  It is carried by flea vectors, but if the dispersal distance of a flea is less than the diameter of a coterie’s territory, then the transmissibility of this vector-borne disease is similar to something that is directly transmitted.  Prairie dogs are territorial and this territoriality limits the rate of disease propagation through prairie dog towns. However, prairie dogs are not alone on their eponymous prairies.

Grasshopper mice – smelly, carnivorous mice, happy to eat through prairie dog carcasses – get swamped by fleas that normally live on prairie dogs. And grasshopper mice have no respect for prairie dog territories. They spread fleas across prairie dog coteries. This is the critical piece of the puzzle provided by our analysis.  Grasshopper mice are the key amplifying hosts for plague in prairie dogs.  Grasshopper mice increase the spread of disease by moving fleas across the landscape, similar to the way that highly promiscuous people may spread HIV or so-called ‘super-spreders‘ transmitted SARS in the global outbreak of 2003.  Of course, there are interesting differences between the plague model and these other diseases. Grasshopper mice are like super-spreaders in that they push the system over the percolation threshold.  They are unlike super-spreaders in that they don’t have that many more contacts than the average – they just connect otherwise unconnected segments of a population already near the threshold of an epidemic.

Without grasshopper mice, plague still kills prairie dog families, one at a time, but it moves very slowly, and it is extremely hard to detect (who misses 5 dead prairie dogs in a colony that stretches for 200 hectares and has upwards of 5000 animals?).  The grasshopper mice take a spatially-organized system that is on the verge of an epizootic and push it over the threshold.  The term ‘percolation threshold’ in the title of our paper relates to a branch of theory from geophysics that explains how and when a fluid can pass through a porous random medium.  This theory uses random graphs, which are the same mathematical structure that we use to model social networks, to understand when, for example, a medium will let water pass through it – i.e., to percolate. When the density of pores in, say, a layer of sandstone passes a critical density, water can pass from the surface through to recharge the aquifer. Similarly, when the density of susceptible prairie dog families crosses a critical threshold, plague can sweep through and wipe out a town of thousands of individuals.  The spatial structure induced by prairie dog territoriality turns out, on average, to be not quite at the percolation threshold (though it’s close).  What the grasshopper mice do is provide the critical connectivity that puts the system over the threshold and allows a slowly simmering enzootic infection turn into a full-blown epizootic.

It is in thinking about percolation thresholds that we see how important the behavior of affected species is for understanding disease dynamics.  Plague in Asian great gerbils, while effectively modeled using the same mathematical formalism, only requires one species in order to achieve the percolation threshold. Because great gerbils roam more widely and mix more, what matters for plague epizootics in this species is simply overall gerbil density.

It seems quite likely that this pattern of diseases smoldering at low-level below the detection threshold before some dramatic occurrence brings them to general attention is common, particularly with emerging infections. For example, there is evidence for extensive transmission of H1N1 ‘swine flu’ in Mexico before a large number of deaths appeared seemingly quite suddenly in April of 2009.  A number of other diseases – both of people and wildlife – show this pattern of being seemingly completely lethal, burning through host communities, and disappearing only to reappear some years later.  Important examples include Ebola in both humans and gorillas, hantavirus in people, anthrax in zebra, or chytrid fungi and frogs.

What are the key take-home messages of this paper? There are five, as far as I see it: (1) plague is enzootic in prairie dogs and there is no need to posit an alternate reservoir, (2) this said, the transition from enzootic to epizootic infection in prairie dogs is mediated by grasshopper mice, (3) understanding disease ecology – including species interactions – is a key to understanding (and predicting) dynamics, (4) behavior matters for disease dynamics, and (5) epidemiological surveillance is essential for controlling infectious disease – just because you don’t see a disease, doesn’t mean it’s not there!

I’m sure I’ll have more to say about this.  I did want to note that the publication of this paper coincides with a personnel transition here in our group at Stanford. Marcel has moved on to a faculty position, joining the spectacular Center for Infectious Disease Dynamics at Penn State.  Peter Hudson and his crew have assembled an amazing and eclectic group of scientists in Happy Valley and kudos to them for landing Marcel.  I frequently think that only a total fool would pass up an offer to join this exciting and productive group, but that’s another story. I expect Marcel to do great things there and look forward to continued collaborations.

New Paper: Dynamics and Control of Diseases in Networks with Community Structure

Marcel Salathé and I have a brand new paper out in today’s issue of the Public Library of Science, Computational Biology. There is also a news piece by Adam Gorlick in the Stanford Report this morning. This is an idea I’ve been bouncing around for a few years now and I was very fortunate to have Marcel – and his programming wizardry – show up with an interest in the very same topic just at the right time. It’s not every day that one of the most talented young theoretical biologists in the world shows up at your office wanting to collaborate. If it ever happens to you, I suggest you act!

The fundamental question is: Does social structure affect that course of epidemics? The answer seems obvious, particularly for infectious diseases that are transmitted by direct person-to-person contact. However, specific work demonstrating the effects of social structure on epidemics can be hard to find. Part of the problem, of course, is that you can hardly do experiments in which you change social structure and then subject populations to an infectious disease. To overcome this ethical and practical barrier to research, epidemiologists, biologists, and social scientists interested in disease and human behavior use mathematical and computational models to study how changes in host behavior affect the outcome of simulated epidemics.

Two specific topics that clearly have some bearing on social structure have been investigated extensively: individual heterogeneity in contact number and individual assortativeness. Epidemic behavior in all but the simplest models has been seen as being driven by heterogeneity. When there is a lot of variance in the number of potentially infectious contacts that individuals in a population have, epidemics are more likely, they infect large segments of the population more quickly, and ultimately infect a larger fraction of the total population. Consider the extreme case where all members of a population have one contact except for one person, who has a contact with everyone else. If we were to draw a picture of such a contact network, it would resemble a star or a wheel with a central hub and spokes:

star

Infect any random individual on this star and everyone else is at risk for infection. At the opposite extreme, if everyone has exactly one contact, then a randomly infected person can infect, at most, one other individual.

couples

Assortativeness, the tendency for individuals to associate with others like themselves, can either aid or hinder the spread of infections. People in contemporary nation states like the United States show an incredible capacity to form associations with like individuals. We form social relationships, particularly intimate relationships, with people who are similar to us in age, socioeconomic status, sexual orientation, ethnicity, education, religion, forms of deviance behavior such as drug use or criminal activity, etc. Frequently, this assortativeness has the effect of localizing and concentrating epidemiologically important contacts. When this happens, individuals who act as bridges between different communities take on central epidemiological importance. For example, married men who visit commercial sex workers can serve as a critical bridge connecting high-risk populations of sex workers and injection drug users with the general population. Similarly, health care workers can bridge hospital populations with the general population, a phenomenon important for the emergence of SARS in 2002. (Note that for epidemiological applications, we call such individuals “bridges” but in other applications we might call them “brokers” or “entrepreneurs,” highlighting the general importance of such ideas for understanding society.) The existence of such social bridges highlights the fact that people can also assort on characteristics that are not visible attributes and this type of assortative behavior can increase connectivity. In particular, if people with few contacts tend to be connected to people with many contacts (as in the case of the star), then such disassortativeness can increase the epidemic potential in a population.

The aggregate effects of individual behavioral decisions can have a profound effect on the shape and composition of human populations, but there is more to human populations than simply individual behavior. For one thing, human populations are characterized by a hierarchical structure: individuals typically belong to households and households are aggregated into communities, which are, in turn, aggregated in towns, states, nations, etc. Naturally, there are cross-cutting ties in such hierarchical organization (much like bridges in individual contact networks). Freudian fantasies of primitive hordes aside, even the largely egalitarian societies of hunter-gatherers are characterized by a hierarchical structuring of families, bands, and tribes. Hierarchical structuring is clearly important for understanding social process in human societies.

So what effect does such community structure have on epidemics? To address this question, Marcel and I combined the formalisms of social network analysis and computational models of epidemics. We already know that heterogeneity in contact number can have profound effects on the outcomes of epidemics and that such heterogeneity can change aggregate social structure in complex ways. To avoid such complications, we generated networks where every individual had the exact same number of contacts. The only thing that varied in these toy networks was the likelihood that any randomly chosen connection between two individuals would be either within or between more or less cohesive subgroups (a.k.a., “communities”). Using metrics derived from Graph Theory, the branch of discrete mathematics that provides the basic tools for Social Network Analysis, we were able to characterize the degree of community structure and relate this to the outcome of epidemics simulated on the resulting networks.

It turns out that community structure has an enormous effect on epidemic outcome. In particular, we found that there is a remarkably abrupt transition from small outbreaks to very large outbreaks as we moved from the most structured populations to more moderately structured ones. Populations characterized by extreme community structure have smaller outbreaks because the infection has a hard time getting out of a community before dying out. As more connections to other communities are made – i.e., the community structure is lessened – there are more opportunities for the infection to escape and affect a larger fraction of the total population. While the result sounds intuitively satisfying after the fact, there was little precedent for expecting such an outcome in the mathematical theory of epidemics. This is because none of the standard metrics of an infectious disease – the basic reproduction ratio, in particular – changed as the populations’ community structure changed.

When we investigated the further structural network correlates of epidemic size, we found that one measure in particular predicted epidemic behavior quite well. This measure, known as “betweenness centrality,” harkens back to previous epidemiological interest in bridging individuals. A person with high betweenness lies on many of the shortest paths that connect all individuals in a network. When a person bridges two distinct subpopulations, he or she typically has high betweenness because all paths from individuals in one cluster have to pass through this person to get to the other cluster, and vice-versa. As a population moves from a condition of very high community structure to a more moderate level, the number of people with high betweenness increases. This highlights a particularly interesting contrast with previous models: epidemics are more likely and larger in populations with highly unequal distributions of contacts on the one hand, but also in populations with more equal betweenness.

With the information that betweenness predicts the extent of epidemic spread in populations with community structure, we sought a means to use such information to design intelligent control measures. How do you find people who have high betweenness? As abstract as the concept of betweenness may seem, it turns out to not be that difficult. We start with an infected person and do standard contact tracing. That is, we ask the index case about his or her contacts. Contact tracing is one of the most important tools in the toolkit of the gumshoe epidemiologist. From the index case’s contacts, we pick a random individual and trace his or her contacts. Picking a random individual from this second generation of contact traces, we simply ask “do you know the index case?” If so, we keep going: trace the contacts of a random contact, ask again if this person knows the index case. When we come to an individual who does not know the index case, we have found our bridge. It is the penultimate person in the chain – the person who links the index case to someone he or she doesn’t know. Basically, we do a “random walk” on the social network looking for people who link otherwise unconnected individuals. When we find the bridge, we vaccinate all of his/her contacts. We call our vaccination algorithm the “Community Bridge Finder” (CBF).

When we vaccinate according to this algorithm, we reduce the final size of the epidemic far more than randomly vaccinating the same fraction of people. More interestingly, CBF also does better than the other vaccination algorithm that uses only local network information typically available to epidemiological investigators. This algorithm, known as the “Acquaintance Method,” vaccinates a randomly selected contact of an index case. The idea behind the acquaintance method is that the contacts of a case are more likely than chance to be highly connected individuals themselves in a population with heterogeneous contacts. That is, given that you have a contact, you’re on average more likely to be connected to a hub than to someone with few connections because hubs simply have more connections.

Of course, the way that we constructed our contact networks, we stacked the deck against the acquaintance method. Remember, everyone has the same number of contacts; what varies is how many contacts are within versus between communities. One of the great limiting factors for progress in social network analysis – and network epidemiology in particular – is the paucity of detailed network data from well-defined human populations. A domain that has garnered a lot of interest recently is the analysis of networks created by social media such as Facebook and Twitter. We used data from Facebook when its use was still restricted to particular college campuses to provide networks on which infections could pass. Facebook users typically have many contacts, probably way more than people have in epidemiologically relevant networks. However, because the data come from college acquaintance networks, we were able to prune the networks down toward something hopefully more epidemiologically appropriate. We kept contacts in the networks only if two individuals shared one a several key attributes such as shared dorm or major. What this yielded were a series of networks with heterogeneous contact structure and quite a bit of community structure (the measure of community structure hovered near the values where epidemics transitioned from small to large in our simulated networks). Once again, CBF outperformed the acquaintance method. This provided very strong evidence that community structure really matters for epidemic behavior and that exploiting information on community structure allows us to better control outbreaks of infectious disease.