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Large-scale human epidemics of bubonic plague have been recorded throughout history, from Roman times to the pandemic in the early 1900s. This disease has had a major social and demographic effect7,8,9,10; its arrival in Europe in 1348 led to the death of around one-third of the human population, and even today bubonic plague kills people in many areas of the world1,2,3. Historical data, from a variety of locations, show occasional large outbreaks of plague separated by long disease-free periods, and yet the disease clearly persists7,8,9. Understanding persistence is a common problem in general epidemic modelling11,12,13, and for bubonic plague it is a central historical question. Previous models of bubonic plague have been highly anthropocentric, modelling the disease as if it were transmitted solely within human populations10,14,15. But consideration of the biology shows that bubonic plague is primarily a disease of rodents that is spread by fleas and only occasionally infects humans; such a disease is termed a zoonosis. From this perspective, we formulate an epizootic (animal-based disease) model for the rat and flea populations, and by coupling this with a standard epidemic (human disease) model, we identify epidemic patterns and the circumstances in which the disease causes a large number of human cases.

The life cycle of the plague can be partitioned into four stages. (1) Fleas feeding on an infected rat ingest the bacteria causing bubonic plague, and soon become infectious. (2) When an infected rat dies, its fleas leave to search for a new host. (3) The fleas usually find other rats, infect them, and so spread the disease through the rodent community. (4) Only when the density of rats is low are the fleas forced to feed on alternative hosts such as humans, and a human epidemic occurs. (Although the pneumonic form of the disease can be passed directly from human to human, historical evidence suggests that this is rare8,16, and is only triggered as a consequence of many rat-related cases.) This informal set of rules can be made precise via a set of differential equations, given in the Methods. We identify the potential force of infection for humans, λH, which allows us to assess the expected number of human cases. Owing to the slow localized spread of plague in rodents8,9, the dynamics and persistence of the disease in both humans and rats is appropriately considered using a spatially structured version of this model with stochastic dynamics.

Using a metapopulation framework17,18, the community is divided into a set of subpopulations, each of which has its own stochastic dynamics19 (see Methods). Two phenomena at the local subpopulation scale control the global dynamics of the model (Fig. 1 ). When the proportion of susceptible rats is low (between 25% and 50%), an import of bubonic plague can often lead to highly persistent endemic behaviour in rats and a very low force of infection to humans. In contrast, when the proportion of susceptible rats is high (greater than 80%) the arrival of infection is followed by a short-lived rat epizootic and a large force of infection to humans (Fig. 1). The global persistence of the disease within the metapopulation depends on a small number of rat subpopulations which are in the endemic state and do not cause human cases; these endemic populations occasionally trigger large but short-lived outbreaks in neighbouring rat subpopulations which in turn lead to major human epidemics.

Figure 1: Results from 250 simulations of the stochastic epizootic model of bubonic plague with a rat population of 2,500.
figure 1

a, The probability that an infectious import generates an epidemic/endemic that lasts for more than 1, 2 or 5 years in the rat population. If the disease persists for more than 5 years it is likely to be in the endemic state. b, The potential force of infection for humans over the entire outbreak in rats (∫∞0λH( t)dt) , measured as the total number of infectious fleas that fail to find a suitable rodent host and may therefore bite and infect humans.

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The potential force of infection for humans, λH, from the stochastic metapopulation model can be compared with historical data for mortality in London20 (Fig. 2a and b). Although the predicted number of human cases is erratic with long inter-epidemic periods, plague persists in the rat population (Fig. 2c). In fact, relatively small rodent metapopulations (50,000 individuals) allow the disease to persist globally for many years, although localized extinctions are common. This is attributable to the spatial heterogeneity that develops21 and the nonlinear density-dependent behaviour of the disease in rats. The persistence of the disease within the rodent population presents a new perspective on the historical data, as a sudden outbreak of bubonic plague in humans does not have to coincide with fresh imports of the disease from an external source. This resolves a controversy in the historical records for some European cities during the fifteenth and sixteenth centuries, when despite very tight quarantine controls frequent human epidemics still occurred8,22.

Figure 2: Comparison between historical data and the results from the metapopulation model.
figure 2

a, The annual deaths attributed to bubonic plague in London (1592–1661)20. b and c, Results from a stochastic simulation of 100 years, with a rodent population of 60,000 distributed over 25 subpopulations (see Methods). b, Potential force of infection for humans showing erratic epidemics; for comparison with the historical data, yearly aggregates are shown. c, Number of infectious rats from the same simulation, showing how plague can persist in an isolated but spatially heterogeneous metapopulation.

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Contemporary control or eradication of the plague can be considered using two main approaches, vaccination or culling. However, both methods have associated complications. Because bubonic plague is a zoonosis, primarily passed from rats (via fleas) to humans, there is no ‘herd immunity’ effect23,24 associated with vaccinating humans. For most contagious diseases, once the vaccination coverage has exceeded the vaccination threshold (70–80% for smallpox, 90–95% for measles and 99% for malaria) the disease cannot persist in the population23. For bubonic plague, only those individuals vaccinated are protected, and vaccination of humans cannot provide a means of eradication because the disease is driven by the rat population. An alternative approach is to reduce the rodent population; but our model predicts that there may be difficulties with the timing of any cull. If the rats are kept at a permanently low level, then the risk of a large outbreak in rodents—and therefore the risk of human cases—is reduced. But if the cull is only brought into effect after the first human cases have been reported, then this action can create a far larger force of infection for humans (Fig. 3a). By the time human cases arise, there is already a large reservoir of infection in the rodent population; a cull therefore releases many infected fleas, which may bite humans in the absence of a suitable rodent host. This highlights the importance of considering bubonic plague as an epizootic infection and explicitly modelling the rat dynamics.

Figure 3: Results from the stochastic model for a single rat subpopulation.
figure 3

Solid lines, means; dashed lines, 95% confidence intervals. a, The effect of culling the rats once human cases have been detected. b, The potential force of infection to humans per rat over the entire outbreak shows clear threshold behaviour. The basic reproductive ratio R0 is defined as R0 = βR KF/dF[1 - exp(-aK R)] . Different values of R0 were obtained by varying the flea-searching efficiency, a, and the rat carrying capacity, KR.

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Given the difficulties in control, it is important to calculate the conditions under which a large human outbreak is likely. The recent discovery of antibiotic-resistant strains4 increases the importance of the prevention of human cases in the absence of a reliable cure. For many cities in both developing and developed countries, rats are still a serious problem25 and in many of these areas, including the United States, southern Africa, southern Asia and South America, there are large reservoirs of plague in wild rural rodents26. Hence there exists the potential for the disease to enter the highly susceptible urban rat population, with clear public-health consequences.

From the metapopulation model, the potential number of human cases associated with an outbreak is found to depend primarily on the basic reproductive ratio of the disease in rats, R0 (Fig. 3b), with human cases much more likely above some threshold value. Consideration of the disease in other wild rodent species allows us to place bounds on the flea searching efficiency a, which in turn means that the R0 threshold corresponds to a rat density of about 3,000 km-2. Many urban populations exceed this critical density, and identifying those communities at high risk is of obvious public-health importance.

Methods

Deterministic model

The natural history of the epizootic infection leads to five differential equations describing the dynamics of rats and fleas,

Here SR is the number of susceptible rats, IR is the number of infectious rats, RR is the number of resistant rats and TR = SR+ I R+ RR is the total rat population. In the absence of plague, the rat population experiences logistic growth, with slowly waning resistance to the disease at the population level governed by the parameter f. N gives the average number of fleas on a rat (the flea index), and F is the total number of infected fleas currently searching for a host. N is again modelled by logistic growth, although it is increased whenever a searching flea finds a new host. Given that fleas search randomly for a new rat host in a limited area, the probability of success is given by 1 - exp(-aTR), where a is a measure of searching efficiency.

λH is the potential force of infection to humans, and as such gives a measure of the number of expected human cases. In fact λH is calculated as the number of infected fleas that fail to find a rat host, and therefore is proportional to (but greater than) the rate at which human cases occur. This set of differential equations always shows oscillatory convergence to an equilibrium point with a very low λH.

Parameter values

See Table 1.

Table 1 Parameter values
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Stochastic metapopulation

Stochastic spatial dynamics are becoming an increasingly common form of epidemiological modelling. We move from a deterministic framework to a stochastic (or Monte Carlo) simulation, by allowing each possible event (for example, birth, death, infection) to occur randomly but at a rate given by the differential equations19. To introduce spatial structure, the entire rodent and flea population is discretized into 25 smaller subpopulations in a square lattice arrangement. Each subpopulation has its own stochastic dynamics derived from the differential equations, and adjacent subpopulations are coupled by the random movement of rats and fleas at rates µR = 0.03 and µF = 0.008, respectively. These movement rates are consistent with the observed slow spread of the disease through communities8,9.