Don’t close the gates

It seems obvious that restricting travel should help prevent the surge of epidemics. But a new mathematical analysis now demonstrates that mobility often reduces the heterogeneity in population distributions, thereby lowering the epidemic risk.

Public health authorities — and indeed human instinct — have long taken for granted that restricting mobility can prevent a localized outbreak from growing into an epidemic. This belief has led to calls for international border closure during all of the recent outbreaks, including Zika, Ebola, H1N1 and SARS. Writing in Nature Physics, Gómez-Gardeñes et al.¹ have now challenged the assumption that movement spreads pathogens and thus facilitates epidemics. Surprisingly, their work suggests that mobility might actually lower the risk of widespread epidemics.

Credit: Reticulations. Woodcut print by Betty Scarpino and Samuel Scarpino.

Although it is true that isolating infectious individuals can slow disease transmission — by, say, closing live bird markets during avian influenza outbreaks² — it remains controversial, both in the public health and scientific communities, whether the wholesale restriction of movement can prevent epidemics³. For example, mathematical and computational models of Ebola⁴ and influenza transmission⁵ found that closing international borders might only delay the epidemic peak by a few weeks and would not reduce the total number of cases.

Despite their colloquial use, the distinction between outbreaks and epidemics has a specific definition, which is grounded in the physics of spreading processes⁶. The mathematical model used by Gómez-Gardeñes et al. categorizes individuals as either susceptible to infection or infectious and, critically, as susceptible again after recovering from their infection. Within this framework, known as a susceptible–infected–susceptible (SIS) model, the transition from an outbreak to an epidemic is characterized by the system entering a stable equilibrium with a non-zero fraction of infectious individuals present in the population.

It is this mathematical distinction — between small outbreaks and large epidemics — that Gómez-Gardeñes et al. studied in the context of mobility. Their model considers epidemics as a reaction–diffusion process, where individuals are infected via local interactions with infectious individuals (reactions) and can move throughout a larger metapopulation (diffusion). If you are unfamiliar with the concept of a metapopulation, think of a collection of cities with movement between them for work. Reaction–diffusion models, although most common in chemistry, have also seen widespread application across the physical and natural sciences. In these models, it is often possible, after making a few simplifying assumptions, to construct an analytical approximation for when the sputtering transmission chains of an outbreak will transition into the stability of an epidemic.

Gómez-Gardeñes et al. studied how the epidemic threshold, the critical point where outbreaks transition to epidemics, is affected by mobility. Specifically, the authors provided an analytical approximation to the eigenvalues of a matrix encoding how individuals are connected in the metapopulation. For a collection of cities, this matrix would record the probability that an individual in city i would contact an individual in another city j. The epidemic threshold is related to the largest eigenvalue of this matrix — similar to using the eigenvalues of the Jacobian to study critical transitions. For low values of mobility — namely, when individuals in city/population i are much more likely to interact with each other than with individuals in population j — the authors found that the epidemic threshold increases, which implies a lower probability of an outbreak growing into an epidemic.

The interesting physics in the paper by Gómez-Gardeñes et al. — that mobility hinders epidemic spread — depends on two key assumptions. First, the individuals must recover and again become susceptible to infection. There are many human, animal and plant diseases like this — seasonal influenza and gonorrhoea being two examples in humans⁷. However, it would be interesting to study how mobility affects epidemic risk for diseases where individuals have lifetime immunity after infection — a so-called susceptible–infected–recovered (SIR) model. Second, the population sizes in the metapopulation must be different. In general, greater asymmetry in population sizes leads to higher epidemic risk and, because mobility reduces population size asymmetry, it lowers epidemic risk. Lastly, it is worth remembering that a set of standard assumptions will always apply to using such an eigenvector approach to study critical transitions.

Despite the surprising discovery of Gómez-Gardeñes et al., such a counterintuitive pattern is not completely unknown. For example, it was found that dynamically exchanging social contacts (a process also thought to facilitate disease spread) can actually reduce epidemic potential⁸. More intriguingly, a similar phenomenon was discovered by evolutionary biologists in the early 1990s and was expanded on in recent years^9,10. Although a more restricted model was considered, there it was shown that mobility can impede the evolution of beneficial mutations, a process whose physics are closely related to disease spread¹¹.

Gómez-Gardeñes et al. also performed an empirical analysis of high-resolution mobility data from Cali, Colombia, a city of 2.4 million people, and found mobility rates between neighbourhoods are in the regime where epidemic risk could be lowered by movement. Their finding, that urban metapopulations can have a lower epidemic risk due to mobility, might inform ongoing debates about how urbanization affects the emergence and establishment of disease¹². There are of course broad parameter ranges of their model where mobility does exacerbate epidemic spreading and, rightly, the authors end with a call to action for advancing our empirical understanding of human mobility. Indeed, the epidemiological modelling community is coming to appreciate how complex mobility patterns (for example, in dengue outbreaks in Pakistan¹³) challenge conventional wisdom.

Isolating and caring for infectious individuals is a vitally important public health strategy, which both slows disease spread and reduces morbidity/mortality. However, the paper here provides both analytical and empirical evidence that — as previous studies have suggested about individual diseases — broader mobility patterns can have a range of effects on outbreaks. Indeed, as Gómez-Gardeñes et al. and others have found, challenging long-held assumptions in epidemiology can uncover a richer physics and may ultimately advance our capacity to prevent epidemics.